Category: Uncategorized

$example of converting fractions to rercentages 73% width=$

The word per means ‘out of’ and cent means ‘100’.

Percent means ‘out of 100’.

We can take the
numeratorThe number on top of a fraction, above the dividing line.
of a fraction out of 100 and write it as a percentage.

We basically replace the / ₁₀₀ with a percentage sign, %.

The percentage sign, % is an easier way to write ‘out of 100’.

In this example ⁷³ / ₁₀₀ is written as 73%.

We read this as 73 percent.

Percent means ‘out of 100’.

We can take a fraction out of 100 and write its numerator as a percentage.

$example of writing a fraction as a percentage$

To write a fraction as a percentage we first need to write it as a fraction out of 100.

We write a fraction out of 100 by using equivalent fractions.

This means that we must multiply the
numeratorThe number on top of the fraction.
and the
denominatorThe number on the bottom of the fraction.
by the same amount.

We can first multiply the top and bottom of the fraction ⁴ / ₅ by 2.

After multiplying the top and bottom by 2, we get ⁸ / ₁₀ .

We can now write ⁸ / ₁₀ as ⁸⁰ / ₁₀₀ by multiplying the top and bottom both by 10.

We could have written ⁴ / ₅ as ⁸⁰ / ₁₀₀ in one go by multiplying the top and bottom by 20.

⁴ / ₅ is 80 out of 100, which we can write as 80%.

Supporting Lessons

Fractions to Percentages: Interactive Questions

Fraction to Percent Worksheets and Answers

$fractions to percent worksheet pdf$

$convert fractions to percent worksheet answers pdf$

Converting Percentages to Decimals

Converting Fractions to Percentages without a Calculator

What is a Percentage?

A percentage is a value out of 100. Percent means ‘out of 100’.

We can write the number as a fraction out of 100 or we can use the % symbol. The % symbol is a short way to write ‘out of 100’ as a fraction.

The word ‘percent’ is made up of two parts.

The ‘per’ means ‘out of’ and the ‘cent’ means ‘100’.

In our minds, we can replace the word ‘percent’ with the words ‘out of 100’.

percent means out of 100

For example, 33% just means 33 out of 100. This can also be written as a fraction as ³³ / ₁₀₀ .

Here is 73 percent.

73 percent is 73 out of 100.

73 percent is 73 out of 100 or 73%

We can write this as ⁷³ / ₁₀₀ .

The percent symbol, ‘%’ is often used instead of writing / ₁₀₀ .

We can write 73 out of 100 as 73%. The % sign can be thought of as meaning ‘out of 100’.

$how to convert fractions to percentages example of 73%$

In this example, we have 2 percent.

We can replace the word ‘percent’ with the words ‘out of 100’.

2 percent means 2 out of 100.

$writing a fraction as a percentage example of 2 out of 100, which is 2%.$

We can write this as ² / ₁₀₀ .

We can write this more easily as 2%.

$writing the fraction 2 out of 100 as a percent$

How to Convert a Fraction to a Percentage without a Calculator

To convert fractions to percentages without a calculator, use the following steps:

Multiply the denominator (bottom) of the fraction by a number to make the denominator equal 100.

Multiply the numerator (top) of the fraction by the same number.

Take this new numerator and write a % sign after it.

If the fraction is already out of 100, then we simply write the numerator with a percentage sign after it.

For example, the fraction of ¹² / ₁₀₀ is already out of 100.

$converting a fraction out of 100 into a percentage. 12 percent.$

We simply take the numerator on top of the fraction, which is 12, and write a percentage sign after it.

¹² / ₁₀₀ is 12%.

We will now look at examples of writing a fraction as a percentage without a calculator, where the fraction is not out of 100.

Here we have the fraction ⁸ / ₅₀ .

We can see that it is out of 50, not out of 100.

The first step is to write it as an equivalent fraction out of 100.

$8 out of 50 as an equivalent fraction out of 100$

We need to multiply the 50 to make it 100. We need to figure out how many times 50 divides into 100.

50 × 2 = 100 and so, we multiply the denominator by 2.

$converting fractions to percentages: 8 out of 50 as an equivalent fraction out of 100$

When finding equivalent fractions, we need to multiply the numbers on top and bottom by the same amount.

Because we multiplied 50 by 2, we need to multiply 8 by 2 as well.

8 × 2 = 16 and so our fraction is now ¹⁶ / ₁₀₀ .

$8 out of 50 equals 16 out of 100 as an equivalent fraction$

⁸ / ₅₀ is exactly the same amount as ¹⁶ / ₁₀₀ . We say that these fractions are equivalent.

8 out of 50 equals 16 out of 100 which is 16%

Now that our fraction is our of 100, it is a percentage.

A percentage is just a fraction out of 100.

We can write ¹⁶ / ₁₀₀ more easily at 16%.

We take the numerator of 16 and put a percentage sign after it.

The fraction ⁸ / ₅₀ written as a percentage is 16%.

This also means that 8 is 16% of 50.

$writing the fraction 8 out of 50 as a percentage$

Here is another example of writing 11 out of 25 as a percentage without a calculator.

The first step is to multiply the number on the bottom so that it equals 100. Remember that we need our fraction to be out of 100, not 25.

25 goes into 100 four times and so, we multiply by 4.

$11 out of 25 as an equivalent fraction out of 100$

The next step is to multiply the number on the top of the fraction by the same amount. Because we multiplied the bottom by 4, we will multiply the top by 4 too.

$11 out of 25 as an equivalent fraction out of 100$

The fraction ¹¹ / ₂₅ is equivalent to ⁴⁴ / ₁₀₀ .

¹¹ / ₂₅ is 44%.

$turning fractions into percentages 44%$

In this example we have ⁷ / ₂₀ .

20 goes into 100 five times and so, we multiply the top and bottom of the fraction by 5.

$7 out of 20 as an equivalent fraction$

Multiplying the numerator by 5, we get 35.

⁷ / ₂₀ is the same as ³⁵ / ₁₀₀ .

$7 out of 20 as an equivalent fraction out of 100 and 35%$

The fraction ⁷ / ₂₀ is 35%.

This also means that 7 is 35% of 20.

$writing fractions as percents without a calculator example of 35%$

In this example of turning fractions into percentages, we have ³ / ₁₀ .

$converting fractions to a percentage 3 out of 10 is 30%$

We multiply the numerator and the denominator both by 10 to find an equivalent fraction out of 100.

3 out of 10 as a percentage without a calculator

3 out of 10 is 30 out of 100.

³ / ₁₀ is 30%.

writing 3 out of 10 as a percentage

This also means that 3 is 30% of 10.

So far we have been able to convert fractions into percentages without a calculator because we could use our times tables to make fractions out of 100.

Sometimes it is not so obvious how to make a fraction out of 100. In this case, we can multiply in steps.

Here is ⁴ / ₅ . The first step to turn this fraction into a percentage is to make the number on the bottom equal to 100 by multiplying it.

You may not know how many times 5 goes into 100 and so, we can just make it equal 10 for now.

We can multiply the top and bottom by 2.

$the fraction 4 out of 5 as an equivalent fraction out of 10$

We multiply 5 by 2 and so we multiply 4 by 2.

⁴ / ₅ is equivalent to ⁸ / ₁₀ .

We can now multiply 10 by 10 to make it equal 100.

$writing the fraction 4 out of 5 as a percent using equivalent fractions$

⁴ / ₅ = ⁸ / ₁₀ = ⁸⁰ / ₁₀₀ .

$converting a fraction into a percentage example without a calculator$

⁸⁰ / ₁₀₀ is 80%.

So ⁴ / ₅ is 80%.

$fractions to percents example of 4 out of 5 being 80 percent$

This also means that 4 is 80% of 5.

We can see that if we did this in one step, we still get the same answer.

We turn this fraction into a percentage without a calculator in one step. 100 ÷ 5 = 20 because 10 ÷ 5 = 2.

$4 out of 5 as an equivalent fraction out of 100$

We can multiply 4 by 20 and 5 by 20 to find our equivalent fraction in one step.

We still get 80 out of 100.

4 out of 5 is 80 out of 100 which is 80%

80 out of 100 is 80%.

writing 4 out of 5 as a percentage without using a calculator

Now try our lesson on Converting Percentages to Decimals where we learn how to write a percentage as a decimal.

percentages to decimals

How to Simplify Fractions

How to Simplify Fractions example of 2 sixths simplified to 1 third

To simplify a fraction, divide the
numeratorThe number on top of a fraction, above the dividing line.
and
denominatorThe number on the bottom of a fraction, below the dividing line.
by the same number.

We notice that 2 and 6 are both in the two times table.

Two is the
highest common factorThe biggest number that divides into both numbers. In this case, the biggest number that divides into both two and six.
that divides into both the numerator and denominator.

We divide the top of the fraction by 2 to get '1'.

We divide the bottom of the fraction by 2 as well to get '3'.

The fraction ² / ₆ is simplified to ¹ / ₃.

A fraction is fully simplified if it cannot be simplified any further.

To simplify a fraction, divide the numerator and denominator by the same number.

Simplifying Fractions step by step example

We simplify the fraction ²⁴ / ₆₀ by dividing the
numeratorThe number on the top of the fraction, above the dividing line.
and
denominatorThe number on the bottom of the fraction, below the dividing line.
by their
highest common factorThe biggest number that divides into both numbers. In this case, the biggest number that divides into both 24 and 60
.

The highest common factor is 12, so we will divide the top of the fraction and the bottom of the fraction by 12.

24 ÷ 12 = 2 and 60 ÷ 12 = 5.

The fraction ²⁴ / ₆₀ is simplified to ² / ₅.

Alternatively, we can simplify our fraction in steps by dividing by any number that goes into both numbers.

Both 24 and 60 are in the six times table so we can divide by six first.

²⁴ / ₆₀ simplifies to ⁴ / ₁₀.

However we can see that 4 and 10 are both even and can both be halved.

The fraction is fully simplified to ² / ₅.

We know that the fraction is fully simplified because there is no other number that divides into 2 and 5.

We know this because 2 and 5 are both
prime numbersNumbers that can only be divided exactly by themself and 1. They are not divisible by any other number.
.

Supporting Lessons

Simplifying Fractions: Interactive Activity

Simplifying Fractions Worksheets and Answers

$simplifying fractions worksheet pdf$

$simplifying fractions worksheet answers pdf$

Unit Fractions of Amounts

Simplifying Fractions

What does it mean to Simplify a Fraction?

Simplifying a fraction means to write a fraction as an equivalent fraction with a smaller numerator and denominator. To simplify a fraction, divide the numerator and denominator by the same number. A fraction is fully simplified if it cannot be simplified any further.

For example, we will simplify the fraction ²/₆ .

The numerator is the number on top of the fraction. The numerator is 2.

The denominator is the number on the bottom of the fraction. The denominator is 6.

$denominator and numerator of a fraction$

To simplify a fraction, divide the numerator and denominator by the same number. The division must be exact because we cannot have a decimal number as the numerator or denominator.

2 and 6 are both even so we can divide them both by 2.

$simplifying the fraction of two sixths by dividing by two$

To simplify a fraction, you must divide the numerator and denominator by the same value.

Since we have divided the numerator by 2, we also need to divide the denominator by 2.

example of simplifying two sixths to one third

²/₆ simplifies to the fraction ¹/₃ .

A fraction written in its simplest form means that it cannot be simplified any further. This means that there is no equivalent fraction with a smaller numerator and denominator.

We cannot divide 1 and 3 by another number exactly. This means that the fraction of ¹/₃ is fully simplified.

The process of simplifying a fraction is also known as reducing a fraction.

Reducing a fraction to its lowest terms means that the fraction has been simplified fully. A fully reduced fraction does not have an equivalent fraction with a smaller numerator or denominator.

How to Simplify Fractions Step-by-Step

To simplify a fraction fully, follow these steps:

Write all of the factors of the numerator and the denominator in two separate lists.

Divide the numerator by the largest number to appear in both lists.

Divide the denominator by the same number.

For example, fully simplify the fraction ²⁴/₆₀ .

$example of simplifying the fraction 24 out of 60$

The first step is to list all of the factors of the numerator and then list all of the factors of the denominator.

highest common factor of 24 and 60 by listing factors of both numbers to get a hcf of 12

The largest number in both lists is 12. 12 is the highest common factor (greatest common factor) of 24 and 60.

The highest common factor of two numbers is the largest number that can divide exactly into both numbers.

The next step is to divide the numerator by the highest common factor.

24 ÷ 12 = 2 and so, our new simplified numerator is 2.

$simplifying 24 out of 60 with equivalent fractions$

The final step is to divide the denominator by the highest common factor.

60 ÷ 12 = 5 and so, our new simplified denominator is 5.

$fully simplifying the fraction 24 out of 60 written in its lowest terms as 2 fifths$

²⁴/₆₀ fully simplified equals ²/₅ .

To reduce a fraction to its lowest terms, divide the numerator and denominator by the greatest common factor.

No number, other than 1, will divide into both 2 and 5, so we have fully simplified the fraction into its simplest form.

The denominator is '5' which is a prime number.

We know a fraction is fully simplified if both the numerator and denominator are prime numbers. A fraction is fully simplified if the denominator is a prime number and the numerator is a smaller number than this. These are not the only ways to know if a fraction is fully simplified but they are two useful checks.

We can also see that the numerator can only be divided by 2 and that the denominator of '5' is odd and therefore cannot be divided by 2.

Why do we Simplify Fractions?

Fractions are simplified to make calculations easier. The simplified fraction is the same value as the original fraction but it has smaller numbers. Smaller numbers are easier to work with.

For example, here is the fraction ¹⁵⁰/₃₅₀ .

If you had to multiply or add this fraction to another, the numbers would be quite large and could take longer to work out than if looking at smaller numbers.

A fraction does not need to be fully simplified in one step. Several steps can be used to fully simplify a fraction by dividing by any factor of both the numerator and the denominator each time.

We can see that both 150 and 350 end in a zero. This is a big clue that both numbers are divisible by 10.

We will simplify the fraction ¹⁵⁰/₃₅₀ to ¹⁵/₃₅ by dividing the numerator and denominator by 10.

$reducing the fraction 150 out of 350$

The fraction has been reduced but it can be reduced further with another step.

Both 15 and 35 end in a 5, which tells us that both numbers are divisible by 5.

We will simplify the fraction ¹⁵/₃₅ to ³/₇ by dividing the numerator and denominator by 5.

$simplifying a fraction step by step$

We can see that the fraction is now reduced to its lowest terms because both 3 and 7 are prime. They cannot be divided by a number apart from one and themselves. There is no number that divided exactly into both 3 and 7 to make them smaller.

We can see that the fraction ³/₇ is much simpler to use than ¹⁵⁰/₃₅₀ .

We reduce fractions to their lowest terms because it is easier to appreciate their size and compare them. It is also easier to use reduced fractions in calculations.

For example ³/₇ is easier to visualise than ¹⁵⁰/₃₅₀ even though they represent the same number.

How to Simplify a Mixed Number

To simplify a mixed number, follow these steps:

Leave the whole number part the same.

Find the greatest common factor of the numerator and the denominator.

Divide the numerator and denominator by the greatest common factor.

Write the simplified fraction immediately after the whole number part.

To simplify a mixed fraction, only simplify the fraction part. Leave the whole number part the same.

For example, simplify 5 ²/₄ .

The first step is to leave the whole number part the same. 5 remains as 5.

We can simplify ²/₄ to ¹/₂ by dividing both the numerator and the denominator by 2.

$how to simplify a mixed fraction$

To simplify the mixed number 2 ⁹/₁₅ , we leave the whole number part the same and simplify ⁹/₁₅ .

We write down the 2 and simplify ⁹/₁₅ to ³/₅ by dividing the numerator and denominator by 3.

The mixed fraction of 2 ⁹/₁₅ simplifies to 2 ³/₅ .

How to Simplify an Improper Fraction

To simplify an improper fraction, simply divide the numerator and denominator by their greatest common factor. Simplify an improper fraction in exactly the same way as simplifying a proper fraction.

An improper fraction behaves in the same way as a proper fraction. An improper fraction is simply a fraction with a larger numerator on the top than its denominator on the bottom.

For example, simplify the fraction ²⁰/₈ .

The greatest common factor of 20 and 8 is 4. 4 is the largest number that divides exactly into both 20 and 8.

The improper fraction of ²⁰/₈ simplifies to ⁵/₂ when the numerator and denominator are both divided by 4.

$how to simplify an improper fraction$

Here is the example of reducing the improper fraction of ⁴²/₁₈ .

The greatest common factor of 42 and 18 is 6. 6 is the largest number that divides exactly into both 42 and 18.

We divide the numerator and denominator by 6 to simplify ⁴²/₁₈ to ⁷/₃ .

When we simplify an improper fraction, the answer is still an improper fraction.

Now try our lesson on Unit Fractions of Amounts where we learn what unit fractions are and how to calculate unit fractions of amounts.

$unit fractions$

How to Find Equivalent Fractions

Equivalent Fractions: Interactive Activity

$how to make an equivalent fraction to one half$

Equivalent fractions are the same size but are written differently.

To be equivalent fractions, the
numeratorsThe number on the top of the fraction, above the dividing line
and
denominatorsThe number on the bottom of the fraction, below the dividing line
must be multiplied or divided by the same number.

The two fractions: ¹ / ₂ and ² / ₄ are equivalent fractions.

The numerator and denominators were both multiplied by 2.

We can multipliy the numerator and denominator by 2 again to find another equivalent fraction of ⁴ / ₈ .

We can multiply both the numerator and denominator of ¹ / ₂ by 4 to make ⁴ / ₈ .

We can also divide the numerator and denominator of a fraction to make an equivalent fraction.

We could choose to multiply both the numerator and denominator by an infinite range of different numbers to make further examples of equivalent fractions.

To make an equivalent fraction, multiply or divide the numerators and denominators by the same number.

$example of finding an equivalent fraction to one third$

To find an equivalent fraction to ¹ / ₃ , we must multiply both the numerator and denominator by the same number.

In this example, we chose to multiply by 2.

¹ / ₃ and ² / ₆ are equivalent fractions.

We can see that they take up the same fraction of the shaded space in the shape shown.

Supporting Lessons

Finding Equivalent Fractions: Interactive Activity

Finding Equivalent Fractions Worksheets and Answers

$finding equivalent fractions worksheet pdf$

$finding equivalent fractions worksheet pdf$

Equivalent Fractions or Not Worksheets and Answers

$equivalent fractions or not worksheet pdf$

$finding equivalent fractions worksheet pdf$

Equivalent Fractions: Missing Numerator or Denominator

Equivalent Fractions

What are Equivalent Fractions?

Equivalent fractions are fractions that are the same size but have different numerators and denominators. To be equivalent, the numerator and denominator of a fraction must have been multiplied or divided by exactly the same number to make the new fraction.

For example, in the fraction ² / ₃ , the numerator and denominator can be multiplied by 2 to find the equivalent fraction ⁴ / ₆ .

The numerator of 2 has been doubled to get 4 and the denominator of 3 has been doubled to get 6.

Both fractions are equivalent because ⁴ / ₆ is ² / ₃ .

One third of 6 is 2 and two thirds of 6 is 4.

Therefore 4 is two thirds of 6. This means that ⁴ / ₆ is the same as ² / ₃ .

There are an infinite number of equivalent fractions that a fraction can have. This is because we can choose any number to multiply the numerator and denominator by and there are infinite numbers.

For example, this time instead of multiplying the digits in ² / ₃ by 2, we will multiply by 5.

$finding a fraction equivalent to 2/3 which is 10/15$

2 multiplied by 5 equals 10 and 3 multiplied by 5 is 15.

We say that fractions that have the same ratio of numerator to denominator are called equivalent.

² / ₃ is equivalent to ¹⁰ / ₁₅ .

This also means that ² / ₃ is equivalent to ⁴ / ₆ which is equivalent to ¹⁰ / ₁₅ is ² / ₃ .

How to Find Equivalent Fractions

To find equivalent fractions, follow these steps:

Multiply or divide the numerator and denominator by the same number.

Only divide if the numerator and denominator remain as whole numbers.

Foe example, ³ / ₄ is equivalent to ³⁰⁰ / ₄₀₀ .

$example of finding an equivalent fraction to 3/4$

The numerator and the denominator have been multiplied by the same amount, so they are equivalent.

The numerator and denominator have both been multiplied by 100.

3 × 100 = 300 and 4 × 100 = 400.

When teaching equivalent fractions, it is important to state that we only multiply and divide and never add or subtract. It is a common mistake for children to add or subtract because they are often more familiar with these processes than they are with multiplication and division.

We can see this common mistake easily in the following diagram. When teaching equivalent fractions, it can help to show this common mistake using a diagram.

We can see that the first fraction is ¹ / ₂ but by adding 2 to both the numerator and denominator, we get to the fraction ³ / ₄ .

$teaching a common mistake when finding equivalent fractions$

We can see that ³ / ₄ is clearly a larger fraction than ¹ / ₂ and so, the fractions are not equivalent.

Why Does the Rule for Equivalent Fractions Work?

The rule for equivalent fractions works because the numerator and the denominator are both multiplied or divided by the same number. Increasing the numerator increases the size of the fraction and increasing the denominator decreases the size of the fraction.

If the numerator and denominator are multiplied or divided by the same amount, the fraction has increased and decreased by the same amount and so, it has not changed its overall size.

The denominator of a fraction tells us how many parts it is split up into in total.

The numerator of a fraction tells us how many parts we have.

If we double the number of parts there are in total and also double the number of parts we have, the proportion of parts we have has not changed.

For example, in the fraction ¹ / ₂ , we have 1 part shaded out of 2.

$why does the rule for equivalent fractions work$

If we double the number of parts we have in total, then each part is twice as small. We now need two parts to have half of the shape.

¹ / ₂ is equivalent to ² / ₄ . This is because 2 is half of 4.

If we double the number of parts again, each part becomes even smaller. Each part is now four times smaller than the half parts that we had originally.

¹ / ₂ is equivalent to ⁴ / ₈ . This means that 4 is half of 8.

All of these fractions are equivalent because they take up the same space. The diagrams show that they are the same amount of the total shape shaded.

Equivalent Fractions Chart

An equivalent fractions chart is a visual way to show fractions that are equivalent to each other. Fractions that are the same size are shown highlighted on an equivalent fraction chart.

Here is an equivalent fraction chart showing fractions equivalent to ¹ / ₂.

$fractions equivalent to one half shown on a fraction wall$

There are infinite fraction that are equivalent to one half, such as ² / ₄ and ³ / ₆ .

Here is an equivalent fraction chart showing fractions equivalent to ¹ / ₃ .

$fraction wall showing fractions equivalent to one third$

There are infinite fractions that are equivalent to one third, such as ² / ₆ or ³ / ₉ .

Fraction charts are also known as fraction walls and can be a very useful method of teaching the concept of equivalent fractions.

An activity for children can be to look for and identify equivalent fractions using a fraction wall.

Improper Equivalent Fractions

Two Improper fractions can be equivalent too. To find equivalent improper fractions, multiply or divide the numerator and the denominator by the same amount.

For example here is the improper fraction ⁵ / ₂ .

We can multiply the numerator and denominator by any amount to make an equivalent fraction as long as this number is the same.

$improper equivalent fractions$

In this example, we have multiplied the numerator and denominator both by 3.

⁵ / ₂ is equivalent to ¹⁵ / ₆ .

Where are Equivalent Fractions Used in Real Life?

Here are some examples of using equivalent fractions in real life:

Finding amounts of ingredients to use in cooking.
Sharing out food that is cut into slices.
Working out Percentages.

For example, you are following a recipe to make a cake that requires half a cup of water. If the cup you are measuring it in only has measurements in quarters, then you can use equivalent fractions to know that 2 quarters are the same as one half.

If you are sharing a 12 slice pizza between 2 people, then you will know that ¹ / ₂ is equivalent to ⁶ / ₁₂ and so, you get 6 slices each.

Percentages are found in real life in test scores, computer downloads and many other places. Percentages are fractions out of 100. To convert to a percentage, we need to use equivalent fractions to find a fraction out of 100.

For example, if you scored 12 out of 20 in a test and wanted to know what percentage this was, you would use equivalent fractions.

Multiplying the numerator and denominator by 5 would convert ¹² / ₂₀ to ⁶⁰ / ₁₀₀ and so, you would have scored 60%. Equivalent fractions are used in real life whenever we need to convert to a percentage.

Now try our lesson on Equivalent Fractions: Missing Numerator or Denominator where we learn how to find a missing numerator or denominator in an equivalent fraction.

$equivalent fraction with a missing numerator$

Percentage Increase and Decrease Word Problems

Summary Example Video Lesson

Summary Example Video Lesson

Converting Percentages to Fractions

Real life percentage increase question, increasing a chocolate bar by 10%

To find 10%, divide a number by 10.

The original mass of chocolate is 200 grams.

200 ÷ 10 = 10 and so 10% of 200 grams in 20 grams.

To increase an amount by 10%, add 10% to the original amount.

200 + 20 = 220. Therefore the new mass is 220 grams

how to calculate simple percentages summary poster

Add the percentage to the original amount to increase by a percentage.

Subtract the percentage from the original amount to decrease by a percentage.

Real life percentage decrease word question, decreasing the price of jeans by 40%

To find 40%, first find 10% and then multiply it by 4.

10% is found by dividing the number by 10. £50 ÷ 10 = £5 and so, 10% is £5.

We multiply 10% by 4 to get 40%. £5 × 4 = £20 and so, 40% is £20.

In a sale, the price is decreased.

To decrease by a percentage, subtract the percentage from the original number.

£50 – £20 = £30 and so, the new price is £30.

Supporting Lessons

Percentages of Amounts

Percentage Change Word Problems

How to Work out Percentage Change

To work out percentage change, follow these steps:

Work out the percentage by dividing the original number by 100 and multiplying by the percentage.

For a percentage increase, add this percentage to the original number.

For a percentage decrease, subtract this percentage from the original number.

For example, in this percentage change question, we will increase $52 by 23%.

Step 1 is to find the percentage.

To find 23%, divide $52 by 100 and then multiply by 23.

finding a percentage of an amount 23% of $52

$52 ÷ 100 × 23 = $11.96. Therefore 23% of $52 is $11.96

We are going to increase the value by 23%.

Step 2 is to add the percentage to the original number.

increasing an amount of money in dollars by 23%

$52 + $11.96 = $63.96. Therefore $52 increased by 23% is $63.96

Here is another example with percentage decrease. We will decrease $75 by 36%.

finding 36% of $75

The first step is to find the percentage by dividing the amount by 100 and multiplying by the percentage.

$75 ÷ 100 × 36 = 27. Therefore 36% of $75 is $27.

This is a percentage decrease question.

The next step is to subtract the percentage from the original amount.

subtracting a percentage from an amount of money

Here are some common percentages that can be found without a calculator:

To find 1%, divide by 100.
To find 5%, divide by 20.
To find 10%, divide by 10.
To find 20%, divide by 5.
To find 25%, divide by 4.
To find 50%, divide by 2.

Percentage Increase Word Problems

Here is a real life example of word problems involving percentage increase.

Examples of percentages in real life

In this example, we increase the mass of a bar of chocolate by 10%

Real life percentage increase word question, increasing a chocolate bar by 10%

The first step is to work out the percentage value. We want to find 10%.

To find 10% of a number, divide it by 10.

Real life percentage increase worded question, increasing a chocolate bar by 10%

10% of 200 grams is 20 grams.

This is a percentage increase word problem, so we add the percentage to the initial amount.

Real life percentage increase question, increasing a chocolate bar by 10%

200 + 20 = 220 and so, the new mass of chocolate is 220 g.

Percentage Decrease Word Problems

Here is a real life example of word problems involving percentage decrease.

Real life percentage decrease worded question, decreasing the price of jeans by 40%

This is a percentage decrease problem because we want to find the new price of the item after it has gone on sale. A sale means that the price has gone down.

The first step is to find the percentage value.

40% is found by finding 10% and multiplying this by 4.

10% is £5 because £50 divided by 10 = £5.

Real life percentage decrease question, decreasing the price of jeans by 40%

We multiply 10% by 4 to find 40%. 40% is £20.

Real life percentage decrease worded question, decreasing the price of jeans by 40%

This is a percentage decrease question because we want to take 40% off the total price.

In a percentage decrease problem, subtract the percentage from the original amount.

£50 – £20 = £30 and so the sale price is £30.

Real life percentage decrease word question, decreasing the price of jeans by 40%

Now try our lesson on Converting Percentages to Fractions where we learn how to write a percentage as a fraction.

$writing a percentage as a fraction$

How to Calculate Percentages of Numbers

Percentage of a Number Calculator

Enter the percentage and the amount that you wish to calculate the percentage of:

% of

How to Calculate a Percentage of any Number

how to find a percentage of an amount

how to calculate a percentage of a number

To calculate the percentage of a number, multiply the number by the percentage and then divide by 100.

The formula for finding a percentage of a number can be written as Number × Percentage ÷ 100 = Answer.

We do not write the % sign in our final answer.

For example, to find 31% of 94, multiply 94 and 31 and then divide by 100.

94 × 31 ÷ 100 = 29.14 and so 31% of 94 = 29.14.

We divide by 100 because ‘per’ means divide and ‘cent’ means 100. The word ‘of’ means to multiply.

31% of 94 means 31 divided by 100 multiplied by 94.

To find a percentage of an amount, multiply the amount by the percentage and divide by 100.

Rules to Find Simple Percentages

how to find 19% of 55

To find a percentage of a number, multiply the number and the percentage and then divide by 100.

We can use the formula: Number × Percentage ÷ 100 = Answer.

55 × 19 ÷ 100 = 10.45.

19% of 55 = 10.45.

We could also work this percentage out by hand.

19% is made up of 10% + 9%.

To find 10%, divide by 10. 55 ÷ 10 = 5.5.

To find 1% ,divide by 100. 55 ÷ 100 = 0.55.

To find 9%, multiply 1% by 9. 0.55 × 9 = 4.95.

To find 19%, add 10% and 9%. 5.5 + 4.95 = 10.45.

Supporting Lessons

Finding Percentages of Amounts Without a Calculator

Finding Decimal Percentages of Amounts

Finding Multiples of 10%: Interactive Activity

Finding Percentages of a Number: Interactive Activity

Calculating Percentages

How to Calculate Percentages of Numbers

To calculate a percentage of a number, multiply the number by the percentage and then divide by 100. Do not write a percentage sign in your answer. For example, to find 31% of 94, multiply 94 by 31 and then divide by 100. 94 × 31 ÷ 100 = 29.14.

how to calculate any percentage of a number

The word ‘percent’ means to divide by 100. This is because ‘per’ means divide and ‘cent’ means 100.

In maths, the word ‘of’ means to multiply.

So to find a percentage of a number, we multiply them and also divide by 100.

Here is an example of calculating 19% of 55.

example of finding a percentage 19% of 55

To find a percentage of a number, multiply the percentage by the number and then divide by 100.

To find 19% of 55, multiply 19 and 55, then divide by 100.

55 × 19 ÷ 100 = 10.45. Therefore 19% of 55 = 10.45.

This method of finding percentages is best done on a calculator.

How to Calculate a Percentage with a Calculator

If your calculator has a % key, simply type the percentage followed by the % key and multiply this by the number. If there is no % key on the calculator, multiply the percentage value and the number together but then divide by 100.

Here is an example of finding percentages with a calculator. What is 34.7% of 125?

how to find a percentage of a number with a calculator? example of 34.7% of 125

We multiply 125 by 34.7 and then divide by 100.

125 × 34.7 ÷ 100 = 43.375. Therefore 34.7% of 125 = 43.375.

Alternatively, if your calculator has a % button, then we type the following:

34.7% × 125 = 43.375. We do not need to divide by 100 if we have used the % button on the calculator because the % sign does this for us.

The easiest way to find a percentage of a number is to use a calculator. This is because most percentage calculations involve performing a multiplication that is not easy to do without writing down working out.

Formula to Calculate a Percentage of a Number

The formula to calculate the percentage of a number is Number × Percentage ÷ 100 = Answer. For example, calculating 48% of 64.6, using the formula is 64.6 × 48 ÷ 100 = 31.008.

the formula to calculate percentages of a number

The number is 64.6, the percentage required is 48.

We multiply the two values and then divide by 100.

48% of 64.6. example of calculating a percentage of an amount

64.6 × 48 ÷ 100 = 31.008 and so 48% of 64.6 is 31.008.

Here is another example of finding a percentage of an amount using the formula.

What is 208% of 700?

Even though the percentage is larger than 100, the rule still applies. The formula can still be used in the same way.

finding a percentage of an amount question 208% of 700

The formula to find a percentage of an amount is Number × Percentage ÷ 100 = Answer.

The number is 700 and the percentage is 208.

700 × 208 ÷ 100 = 1456 and so, 208% of 700 = 1456.

The formula of Number × Percentage ÷ 100 = Answer is a good way to find a percentage of an amount when a calculator is available and the numbers involved are larger or contain decimals.

Some percentages are easier to find and can be worked out mentally. These are known as simple percentages.

Rules to Find Simple Percentages

The following table shows the rules used to find basic percentages of amounts:

Percentage	Rule
1%	÷ 100
5%	÷ 20
10%	÷ 10
20%	÷ 5
25%	÷ 4
33.3̄%	÷ 3
50%	÷ 2
75%	÷ 4 then × 3

For example, to find 50% of a number, simply divide it by 2.

50% of 60 means 60 ÷ 2.

50% of 60 is half of 60 which is 30

60 ÷ 2 = 30 and so, 50% of 60 is 30.

Here is another example of using the rules of finding simple percentages. We are asked to find 25% of 600.

25% of 600 is one quarter of 600 which is 150

To find 25% of a number, divide it by 4.

We can divide by 4 by halving and halving again. Half of 600 is 300 and then half of 300 is 150.

25% of 600 is 150.

The rules to find basic percentages of amounts allow us to find percentages without using a calculator. When introducing percentages, it is worth learning the rules to find simple percentages before trying to find more complicated examples.

Here is a downloadable poster showing some of the ways to find simple percentages:

How to Calculate Percentage Increase and Decrease

Here is a list of the rules for finding basic percentages:

To find 1%, divide by 100.
To find 5%, divide by 20.
To find 10%, divide by 10.
To find 20%, divide by 5.
To find 25%, divide by 4.
To find 33.3%, divide by 3.
To find 50%, divide by 2.
To find 75%, divide by 4 then multiply by 3.

Finding Multiples of 10%

To find multiples of ten percent, use the following rules:

Percentage	Rule
10%	÷ 10
20%	÷ 10 then × 2
30%	÷ 10 then × 3
40%	÷ 10 then × 4
50%	÷ 10 then × 5
60%	÷ 10 then × 6
70%	÷ 10 then × 7
80%	÷ 10 then × 8
90%	÷ 10 then × 9

We can use these rules to find any multiple of 10 percent.

For example, in this question we are asked, what is 20% of 40?

To find 20%, first find 10% and then double it.

To find 10%, divide by 10.

example of finding a multiple of 10%. 20% of 40

10% of 40 is 4.

The next step is to double 10% to find 20%.

20% is double 10% so 20% of 40 is 8

20% of 40 is 8.

How to Calculate a Percentage of a Number without a Calculator

To calculate a percentage of a number without a calculator, follow these steps:

Separate the percentage into its tens, ones and decimal place value columns.

Divide the number by 10 to find 10% and then multiply this by however many lots of 10% there are.

Divide the number by 100 to find 1% and then multiply this by however many lots of 1% there are.

Divide the number by 1000 to find 0.1% and then multiply this by however many lots of 0.1% there are.

Add the results together.

how to find any percentage of a number guide

We will start by finding a whole number percentage of an amount. In this example we will find 15% of 40 without a calculator.

The first step is to separate 15% into its tens and ones. 15% = 10% + 5%.

example of calculating 15% of 40.

To find 10%, divide by 10. 40 ÷ 10 = 4 and so, 10% of 40 is 4.

To find 5%, divide by 20. 40 ÷ 20 = 2. Alternatively, we can find half of 10%. 5% of 40 is 2.

The final step is to add 10% and 5% to make 15%. 4 + 2 = 6, therefore 15% of 40 is 6.

Here is another example of finding any percentage of a number.

What is 35% of 220?

35% of 220 split into 30% and 5%

The first step is to split 35% into 30% and 5%.

To find 30%, divide by 10 to get 10% and then multiply by 3 to get 30%.

To find 5%, divide by 100 to get 1% and then multiply by 5 to get 5%.

Alternatively, 5% is half of 10%. 10% of 220 is 22 and so, 5% of 220 is 11.

calculations showing how to find a percentage of a number, 35% of 220

The final step is to add the results of 5% and 30% together to find 35%.

30% is 66 and 5% is 11. 66 + 11 = 77 and so, 35% of 220 = 77.

how to find 35 percent of 220

Here is an example of finding a percentage of an amount. What is 12% of 200?

The first step is to write 12% as 10% + 2%.

To find 10%, simply divide the number by 10.

To find 2%, find 1% and double it.

To find 1%, divide the number by 100.

how to work out 12% of 200 without a calculator

200 ÷ 10 = 20 and so, 10% of 200 is 20.

200 ÷ 100 = 2 and so 1% of 200 is 2.

Double 1% to find 2%. 2 × 2 = 4 and so 2% of 200 is 4.

The final step is to add 2% and 10% to find 12%.

20 + 4 = 24, therefore 12% of 200 = 24.

In this next example of finding a percentage without a calculator, we have 83% of 1200.

We separate 83% into 80% and 3%

calculating 83% of 1200 by calculating 80% and 3%

To find 80%, find 10% and multiply it by 8.

To find 3%, find 1% and multiply is by 3.

calculating 83% of 1200 by calculating 80% and 3%

1200 ÷ 10 = 120 and so, 10% = 120.

80% = 10% × 8 and so, 80% is 120 × 8. 120 × 8 = 960.

1200 ÷ 100 = 12 and so, 1% = 12.

3% = 1% × 3 and so, 3% is 12 × 3. 12 × 3 = 36.

example of 83% of 1200 when calculating a percentage of a number by hand

The final step is to add the results together. 80% + 3% = 83%.

80% = 960 and 3% = 36, therefore 960 + 36 = 996.

83% of 1200 = 996

calculating 83% of 1200 by calculating 80% and 3% which equals 996

When teaching finding percentages of numbers, it is useful to write down the values of all new percentages found as part of the working out. For example, if you have already found 10%, then write this down because we simply need to double it if we need 20% or halve it if we need 5%.

Examples of Calculating Percentages of Numbers

Here are some further examples of calculating percentages of numbers. In these questions, we will work out the percentages without a calculator.

We will work out how to find the percentage of an amount using examples containing decimal percentages.

What is 23.1% of 2000?

Firstly, separate 23.1% into 20% + 3% + 0.1%.

example of finding a decimal percentage of an amount

10% is found by dividing the number by 10. 10% = 200.

20% is double 10%. 20% = 400.

1% is found by dividing the number by 100. 1% = 20.

3% is three times 1%. 3% = 60.

0.1% is found by dividing the number by 1000. 0.1% = 2.

We can add up 20% + 3% + 0.1% to find 23.1%.

400 + 60 + 2 = 462 and so 23.1% of 2000 = 462.

In this next question, we will find a decimal percentage of an amount of money.

What is 47.2% of 400?

To find 47.2%, add 40%, 7% and 0.2%.

finding a percentage of an amount of money example of 47.2% of 400 pounds

To find 10%, divide by 400 by 10. 10% of 400 is 40. To find 40%, multiply 10% by 4. 40% = 160.

To find 1%, divide 400 by 100. 1% of 400 is 4. To find 7%, multiply 1% by 7. 7% = 28.

To find 0.1%, divide 400 by 1000. 0.1% of 400 is 0.4. To find 0.2%, multiply 0.1% by 2. 0.2% = 0.8.

We add 40%, 7% and 0.2% to find 47.2%.

160 + 28 + 0.8 = 188.8. Therefore 47.2% of £400 = £188.80. We write two decimal places with answers involving money.

Here is a question involving finding a percentage of an amount of money in dollars.

What is 84.5% of $1200?

We split 84.5% into 80% + 4% + 0.5%.

finding a percentage of an amount of money example of 84.5% of $1200

To find 10%, divide $1200 by 10.

10% = 120. We multiply this by 8 to get 80%. 80% = 960.

To find 1%, divide $1200 by 100.

1% = 12. We multiply this by 4 to get 4%. 4% = 48.

To find 0.1%, we divide $1200 by 1000.

0.1% = 1.2. We multiply this by 5 to get 0.5%. 0.5% = 6. We can also see that 0.5% is half of 1%. 1% = 12 and so, 0.5% is half of 12, which is 6.

Finally, we add 80% + 4% + 0.5% to get 84.5%.

960 + 48 + 6 = 1014. Therefore 84.5% of $1200 = $1014.

Now try our lesson on How to Calculate Percentage Increase and Decrease where we learn how to find percentage increase and decrease.

percentage change real life examples

How to do Column Addition

how to teach addition without regrouping

To add 32 and 15 we line up the individual digits in the tens and ones columns.

We start on the right by adding the ones together.

2 + 5 = 7.

We now add the digits in the tens column.

3 + 1 = 4.

32 + 15 = 47.

Addition without regrouping

Line up the digits above each other.

Add the digits separately from right to left.

Column Addition with Regrouping

Column Addition of 2-Digit numbers with regrouping

To add 38 and 16 we separate the numbers into their tens and units.

We start on the right by adding the units together.

8 + 6 = 14.

We only write one digit per column in our answer.

We regroup the 14 units into 1 lot of 10 and 4 units. We carry the 1 ten over to the tens column.

There are 4 units remaining.

We can now count number of groups of ten that we have in total.

We have 3 plus 1, plus the extra 1 ten that we carried, which makes 5.

5 tens and 4 units makes 54 in total.

addition with regrouping

Whenever the digits add to make 10 or more, carry the tens over to the tens column.

Column addition with regrouping example of 27 + 48

We line up the digits above each other and add them from right to left.

7 + 8 = 15.

From the 15, we carry the 1 ten and write down 5 in the ones column.

2 + 4 + the 1 we carried = 7.

27 + 48 = 75.

Supporting Lessons

Column Addition Worksheets and Answers

2-digit column addition worksheet answers pdf

2-Digit Column Addition: Interactive Questions

Printable Column Addition Workbooks

Click here to view our Column Addition Workbooks for lots more extra addition practice!

2-Digit Column Addition: Interactive Question Generator

Column Addition

What is Column Addition?

Column addition is the most commonly used written method for adding two or more numbers. The numbers are written with the digits lined up above each other to make columns. The digits in each column are added one at a time from right to left with the answer written below.

How to Set Out Column Addition

To set out column addition, follow these steps:

Write the numbers above each other, lining up the digits in each place value column.

Draw 2 horizontal lines underneath these numbers to create the answer space.

Add the digits in each separate column starting with the rightmost column and finishing with the leftmost column.

Only write the ones digit of your answer below each column.

Write the tens digit of your answer below the answer space of the next column to the left.

It helps to use square grid paper when doing column addition because you can line up the digits of each number more carefully by writing each digit in each grid.

For example here is the column addition of 32 + 15.

32 + 15 set out in column addition

We line up the digits of each number according to their place value columns.

We work from right to left. So, we begin by adding the digits in the units column.

adding the units in the column addition of 32 + 15

2 + 5 = 7

We write our answer in the units column, underneath the 2 and the 5.

Next, we add the digits in the tens column.

adding the tens in the column addition of 32 + 15

3 + 1 = 4

We write the answer in the tens column, underneath the 3 and the 1.

We read the answer to the column addition from between the two answer lines.

Therefore,

32 + 15 = 47

Here is the complete example showing the written column addition method as an animation.

Column Addition 2-Digit numbers example

Column Addition Without Regrouping

To add numbers without regrouping, firstly line up the digits of each number above each other. Add each column of digits separately. Write the answer to the addition below each column.

Regrouping is needed when the sum of digits in each column makes 10 or more. Addition without regrouping is when the digits in each column we add make an answer that is less than 10. This means that we can simply add the digits column by column and write the answer directly below.

For example here is 74 + 24.

We set out the addition like so, with the digits lined up above each other. It can be helpful to use square grid paper to assist with this layout.

Column Addition of 74 plus 24 example

We start at the right and add the digits.

4 + 4 = 8

7 + 2 = 9

And so, 74 + 24 = 98.

Here is another example of column addition without regrouping.

In this example we want to add 53 + 31.

Column Addition 2-Digit 4.gif

Starting by adding the digits in the units column:

3 + 1 = 4

Then moving to the tens column:

5 + 3 = 8

And so, 53 + 31 = 84.

Column Addition with Regrouping

Column addition with regrouping occurs when the sum of any column is 10 or more. We only write down the units digit and carry the tens digit over to add to the next column on the left.

The rule of column addition is that we only write one digit in each answer box.

In regrouping questions, there are too many digits in the answer space for each place value column.

For example, here is 38 + 16.

38 + 16 with 38 shown as counters in the tens and units columns

The number 38 has 8 units, which are shown by the eight counters in the units column.

It also has 3 tens, which are shown by the three groups of ten counters in the tens column.

38 + 16 shown as counters in the tens and units columns

The number 16 has 6 units, which are shown by the six green counters in the units column.

It also has 1 ten, which is shown by the one group of ten green counters in the tens column.

First we, line up the digits of each number in their place value columns. Then we add the digits in each column.

counting the units in the column addition of 38 + 16 using counters

We have a total of 14 units in the units column. The number 14 has two digits.

We can’t write more than one digit in each place value column.

We therefore need to regroup some counters to carry over to the tens column.

regrouping ten units in the addition of 38 + 16 to carry over to the tens column

In the tens column, we count the groups of ten.

Therefore, we can regroup ten counters from the units column and carry them over to the tens column.

38 + 16 shown with counters with the tens carried over

Now that we have carried 1 lot of ten over to the tens column, we are left with 4 units in the units column.

the column addition of 38 + 16 shown with counters

Altogether, there are 5 tens in the tens column, including the ten that we regrouped and carried from the units column.

Therefore,

38 + 16 = 54

Column Addition 2-Digit with carrying or regrouping

When the digits add to a number that is 10 or greater, it is necessary to do the regrouping process.

The ‘1’ is carried over to the next place value column and included with those numbers when the tens are added.

This process is called regrouping or carrying.

We’ll look at this example again, but this time we will use the formal written addition process.

38 + 16 set out as a column addition

First, line the digits up according to their place value columns.

We add columns from right to left. So, we begin with adding the digits in the units column.

looking at the units in the column addition of 38 + 16

8 + 6 = 14

The number 14 has 4 units, so we write 4 in the units column.

It also has 1 ten, so we carry ten over to the tens column and write it below the line.

Next, we add the digits in the tens column. We must also add the 1 ten that we carried over to the tens column.

38 + 16 shown as a column addition

3 + 1 + 1 = 5

Therefore,

38 + 16 = 54

Here is the complete method for adding the numbers shown on paper.

Column Addition 2-Digit 6

It is best to teach the column addition process without carrying before introducing problems involving carrying.

How to do Column Addition

To do column addition, follow these steps:

Write the numbers directly above each other so that the digits in each place value column are lined up.

Add the digits in the ones column.

If this sum is 10 or greater, only write the ones digit of the answer and write the tens digit below the answer lines in the next column on the left.

Add the digits in the next place value column to any tens digits that you carried over from the previous step.

Repeat steps 3 and 4 until all place value columns have been added.

For example, 47 + 25.

Step 1 is to write the numbers directly above each other, lining up the digits.

We write the 7 above the 5 and the 4 above the 2.

Column Addition 2-Digit with carrying example of 47 + 25

Step 2 is to add the digits in the ones column.

7 + 5 = 12

Step 3 is to write the ones digit of the answer and write the tens digit of the answer below the lines in the next column along.

12 contains two digits and so the ‘2’ digit is written and the ‘1’ is carried over to the tens column.

Step 4 is to add the digits in the tens column to any digits that were carried over in the previous step.

In the tens column we have 4 + 2.

We also have the ‘1’ ten that we carried over from our previous answer of 12.

4 + 2 + 1 = 7.

And so, 47 + 25 = 75.

It is a common mistake to forget to add the ‘1’ that has been carried and this is the first place to look for mistakes in column addition if the original procedure without carrying has been mastered.

The next example of column addition with carrying is 34 + 59.

Column Addition example with carrying / regrouping 34 + 59

Adding the digits in the units column we have:

4 + 9 = 13

We need to carry the ‘1’ over to the tens column and simply write ‘3’ in the units column.

Adding the digits in the tens column, we need to remember to add the ‘1’ that has been carried over.

3 + 5 + 1 = 9

Step 5 is to repeat steps 3 and 4 until all columns have been added.

We have already added all of the numbers and so we can now read our answer.

34 + 59 = 93.

Here is another example of column addition with carrying.

We have 27 + 48.

Column Addition 2-Digit addition with regrouping. 27 + 48

7 + 8 = 15.

We write down the ‘5’ and carry the ‘1’.

We need to add this ‘1’ to the numbers in the tens column.

2 + 4 + 1 = 7

And so, 27 + 48 = 75.

Teaching Column Addition

Column addition is taught in primary schools as the most common method for adding numbers. Lots of practice is recommended with using this written method. It helps to use counters or base ten blocks to introduce the idea of regrouping.

To add numbers using the column addition method we line the digits up in their place value columns and add the digits from right to left.

In the column addition example below, we are asked to calculate 32 + 15. We begin by partitioning the digits into their tens and units.

When first introducing the column addition method, it can be helpful to use counters to visualise the sizes of the numbers being added and to explain the carrying / regrouping process.

32 is partitioned into its tens and units columns and is shown as counters

The number 32 has 2 units. These are shown by the two counters in the units column.

It also has 3 tens. These are shown by the three groups of ten counters in the tens column.

We want to add 15.

2-digit column addition of 32 + 15 explained with counters in the tens and units columns

The number 15 has 5 units. These are shown by the five green counters in the units column.

It also has 1 ten. This is shown by the one group of ten counters in the tens column.

Now that we have lined up the digits of each number in their place value columns, we can add the digits in each column.

column addition of the units in 32 + 15 with counters representing the total number

Altogether, in the units column, we have the 2 counters from the 32 and the 5 counters from the 15.

2 + 5 = 7

This gives us a total of 7 counters.

Adding the tens in 32 + 15 with counters representing the numbers

Altogether, in the tens column, we have 3 groups of ten counters from the 32 and 1 group of ten counters from the 15.

3 + 1 = 4

This gives us a total of four groups of ten counters.

Therefore

32 + 15 = 47

Here is the full animated example.

Column Addition of 2-digit numbers without carrying

Now try our lesson on 3-Digit Column Addition without Carrying / Regrouping where we learn how to add three-digit numbers using column addition.

3-Digit Column Addition without Carrying / Regrouping

column addition 3 digit.png

How to Find the Area of a Triangle

example of how to find the area of a triangle

The area of a triangle is one half times base times height.

The area formula can be written as ¹ / ₂ × base × height.

The base and the height must be at right angles to one another.

Here the base is 8 cm and the height is 3 cm.

The area is ¹ / ₂ × 8 × 3 = 12 cm².

The units of area are measured in units squared.

Since the sides are measured in cm, the area is measured in cm².

The area of a triangle is ¹ / ₂ × base × height.

The units of area are measured in units squared.

example of how to find the area of a triangle

The area of a triangle is ¹ / ₂ × base × height.

The base is 11 m and the height is 8 m.

We can multiply the base and height and then halve the answer.

11 × 8 = 88 and half of this is 44.

The area of the triangle is 44 m².

We could have also halved the 8 first and then multiplied by 11.

Half of 8 is 4 and then 4 × 11 = 44.

The units are m² because the side lengths are measured in metres.

Supporting Lessons

Area of a Triangle Worksheets and Answers

area of a triangle worksheet answers pdf

How to Find the Area of a Parallelogram

Finding the Area of a Triangle

How to Find the Area of a Triangle

To find the area of a triangle, multiply the base by the height and then divide by 2. Alternatively, divide the base by 2 and then multiply by the height or divide the height by 2 and then multiply by the base.

It does not matter whether you multiply the base and height first and then halve the answer or whether you halve the base or height first and then multiply.

For example, here is a right-angled triangle of base 8 cm and a height of 3 cm.

area of a right angled triangle

Multiplying the base and height, 8 × 3 = 24. Halve of this answer is 12.

The area is 12 cm².

The area of a triangle is measured in units squared. Take the unit that the sides are measured in and write this unit squared after your answer. If the sides are measured in cm, then the area units are cm². If the sides are measured in m, then the area units are m².

We have multiplied cm by cm in this calculation and so, the area units in this example are cm².

Instead of multiplying the base by the height first and then dividing by 2, we can halve one of the base or height lengths first.

We will look at the same example and divide by 2 first before multiplying.

example of calculating the area of a triangle

Since 8 cm is even and 3 cm is not, it is easier to halve the 8 cm length.

Half of 8 is 4 and then 4 × 3 = 12. The area is 12 cm².

The answer is the the same no matter in what order the numbers are multiplied.

The formula for the area of a triangle is ¹ / ₂ × base × height. This formula can be more easily written as Area = ¹ / ₂ bh.

The formula of Area = ¹ / ₂ bh works for all triangles, no matter what size or shape. As long as the height and base are known, this formula can be used to calculate the area.

Here is an example of using the formula to calculate the area of a triangle.

The base of this triangle is 11 m and the height of the triangle is 8 m.

The value of b = 11 and the value of h = 8.

using the formula to calculate the area of a triangle

¹ / ₂ × b × h becomes ¹ / ₂ × 11 × 8.

We can multiply 11 by 8 first and then halve the answer. 11 × 8 = 88 and then 88 ÷ 2 = 44.

The side lengths are measured in metres and so, the area of this triangle is 44 m².

When teaching finding the area of a triangle, it is recommended to divide by 2 first before multiplying in order to make the calculation easier.

It is easier to halve the 8 first to get 4 and then multiply this by 11 to get the same answer of 44 m².

By doing the division first, the multiplication sum uses smaller numbers which are easier to calculate. This method also involves halving a smaller number rather than the larger final answer.

When using the formula to find the area of a triangle, it is important to ensure that the base and height meet at right angles.

Sometimes you may be given all three sides of a triangle. To calculate the area of a triangle, choose two lengths that are at right angles to each other as the base and height. Multiply these lengths and then divide by 2. Ignore any other side lengths.

Why is the Area of a Triangle One Half Base Times Height

Any triangle can be drawn inside a rectangle that is twice its size. The area of a rectangle is base times height and so, the area of the triangle is one half times base times height.

The area of a rectangle is base × height, alternatively written as length × width.

the area of a rectangle is length times width

To make a rectangle into a triangle, divide it in half diagonally.

why is the area of a triangle half of a rectangle

Since the area of a rectangle is base × height, the area of a triangle is ¹ / ₂ × base × height.

Here is another example of a triangle that is half the size of a rectangle.

area of a triangle inside a rectangle

The area of the rectangle is 10 × 6 = 60 mm² and the area of the triangle is half of this.

The area of the triangle is 30 mm²

Remember that when teaching the area of a triangle, it is easiest to divide either the height or base by 2 first.

Half of 10 is 5 and then 5 × 6 = 30 mm².

example of calculating the area of a scalene triangle

Now try our lesson on How to Find the Area of a Parallelogram where we learn how to find the area of a parallelogram.

Sharing in a Ratio: Bar Model

example of a sharing in a given ratio bar model problem

We can use the bar model to show our given amount as a rectangular bar.

In this example, the whole bar is worth 120.

We can divide the bar into the total number of parts in the ratio.

There are 3 + 1 = 4 parts in the ratio, each worth an equal amount.

120 ÷ 4 = 30.

Each part of our bar model is worth 30.

3 of the parts are worth 90 in total.

The remaining part is worth 30.

120 shared in the ratio of 3:1 is 90:30.

The complete bar in the bar model represents the total amount and we divide it into equal parts to show the proportions in our ratio.

example of sharing in a ratio problem using the bar model

In this example we will share 42 grams of sugar in the ratio 3:4 using the bar model.

The whole bar represents 42 g.

There are 3 + 4 = 7 parts in the ratio in total.

We divide the bar into 7 equal parts.

42 g ÷ 7 = 6 g.

The 3 parts in our ratio are worth 3 x 6 g = 18 g

The 4 parts in our ratio are worth 4 x 6 g = 24 g.

42 g shared in the ratio 3:4 = 18:24.

Supporting Lessons

Sharing in a Given Ratio Worksheets and Answers

shring in a given ratio bar model worksheet pdf

shring in a given ratio bar model worksheet answers pdf

How to Share an Amount in a Given Ratio using a Bar Model

In this lesson we will learn how to share an amount in a given ratio using the bar model.

The bar model is a simple method of showing proportion visually and introducing sharing in a ratio. A rectangular bar is drawn to represent the total amount and it is divided into equal parts in accordance with the given ratio.

Here is our first ratio problem: Share 120 blocks in the ratio 3:1

We are asked to share 120 blocks of chocolate in the ratio 3:1.

There are two numbers in our ratio problem, ‘3’ and ‘1’. This means that we are sharing the 120 blocks between two people.

This means that for every three parts one person is given, the second person receives one part.

We will show the three parts in pink and the one part in blue on our rectangular bar model.

Share 120 blocks in the ratio 3:1 using the bar model

The complete rectangle in the bar model above represents all 120 blocks of chocolate.

The bar model will be divided into equal parts.

There are 3 parts for one person and 1 part for the other person. This is four parts in total.

This means we divide the bar into 4 equal parts.

Share 120 blocks in the ratio 3:1 using a bar model for ratio

Three parts of our bar will be given to one person. These are shown in pink.

One part of the bar will be given to the second person. This is shown in blue.

We need to find out how much each part is worth.

Share 120 blocks in the ratio 3:1 on the bar model by dividing 120 into 4 parts

We must share the 120 blocks over the four equal parts using division.

Because we want to share the blocks equally, we will divide 120 by 4.

120 ÷ 4 = 30

Therefore, each of the four parts on the bar model are worth 30 blocks.

Share 120 blocks in the ratio 3:1 by dividing 120 into 4 parts giving 30 in each part on a bar model

We will now look at the three pink parts to find out how many blocks of chocolate the first person will receive in total.

Share 120 blocks in the ratio 3:1 by dividing 120 into 4 parts to get 90 to 30

We have three parts, each worth 30.

30 x 3 = 90

The first person therefore receives 90 blocks of chocolate.

Share 120 blocks in the ratio 3:1 by dividing 120 into 4 parts to get 90 to 30

The second person receives one lot of 30.

So, they receive 30 blocks of chocolate.

Share 120 blocks in the ratio 3:1 by dividing 120 into 4 parts to get 90 to 30

The chocolate has been shared in a ratio of 90:30.

When teaching ratio problems for the first time, the bar model can be an easy way to show the situation visually.

Now try our lesson on Solving Ratio Problems where we learn the method for solving ratio problems.

Solving Ratio Problems

solving ratio problems

Introduction to Ratio

Introduction to Sharing in a Ratio Summary

We are sharing 35 blocks of chocolate in the ratio 4:3.

There are two numbers in this ratio, so we are sharing the chocolate between two people.

One person gets 4 parts for every 3 parts that the other person gets.

We break off groups of 4 blocks and groups of 3 blocks until there is nothing left.

The total on the left is 20 blocks and the total on the right is 15 blocks.

20 + 15 = 35, which is the total amount of blocks that we started with.

The numbers in a ratio tell us how to share our total amount.

Introduction to Ratio Example

There are 16 blocks in total that will be shared between two people in the ratio 1:3.

This means that every time we give the person on the left 1 block we give the person on the right 3 blocks.

We can share the chocolate in this proportion until it is all gone.

The person on the left has 4 blocks and the person on the right has 12 blocks.

We write the answer as a ratio of 4:12.

We can check our answer since 4 + 12 = 16, which is the total amount we started with.

Supporting Lessons

Introduction to Sharing in a Ratio Worksheets and Answers

introduction to sharing in a ratio worksheet pdf

Sharing in a Ratio using the Bar Model

Introduction to Sharing in a Ratio

A ratio tells us how a total amount has been broken up or can be broken up. In this lesson, we will be sharing a total amount between people. The amount of numbers in the ratio tell us how many people we will be sharing between.

We use a colon ‘:’ to separate the numbers in a ratio and this is a way to help us recognise that we have a ratio rather than a list of numbers.

In the image below, we can see an example of a ratio with two numbers: ‘4’ and ‘3’. Because there are two numbers, we will be sharing the amount between two people.

introduction to what ratio means

The numbers of ‘4’ and ‘3’ tell us the proportions (or relative sizes) of how we will share the amount.

This ratio means that for every 4 parts we give to the person on the left, we give 3 parts to the person on the right.

35 blocks of chocolate in a rectangle 7 blocks by 5 blocks

In our first example of sharing in a ratio above, we have a bar of chocolate made up of 35 blocks in total.

sharing 35 blocks of chocolate in the ratio 4:3

We are asked to share the total amount of 35 blocks in the ratio 4:3.

This means that for every 4 parts one person is given, another person is given 3 parts.

In this introduction to ratio, we imagine that we place the blocks for the first person in a purple pot, and the blocks for the second person in a green pot.

sharing 35 blocks of chocolate in the ratio 4:3

For every 4 blocks of chocolate we place in the purple pot, we place 3 blocks in the green pot.

sharing 35 blocks of chocolate in the ratio 4:3

We can share this ratio by breaking off 4 parts and placing them in the purple pot and then breaking off 3 parts and placing them in the green pot.

Every time we move 4 blocks over to the left, we must move 3 blocks over to the right.

This will ensure that we keep the proportion of 4 parts to 3 parts.

sharing 35 blocks of chocolate in the ratio 4:3 by breaking off blocks of 4 and 3

We continue until we have shared all of the chocolate.

sharing 35 blocks of chocolate in the ratio 4:3 to get 20 to 15

If we count the blocks of chocolate, we can see that the first person receives 20 blocks and the second person receives 15 blocks.

sharing 35 blocks of chocolate in the ratio 4:3 to get 20 to 15

This gives us a ratio of 20:15.

We can check our answer since 20:15 simplifies to 4:3, which was our original ratio that we were asked to share in. We simply divide 20 by 5 to get 4 and divide 15 by 5 to get 3.

An equivalent ratio is a multiple of another ratio. All numbers in the ratio have been multiplied by the same number.

The ratio of 20:15 is an equivalent ratio to 4:3 because both numbers have been multiplied by 5.

We also see that 20 + 15 = 35, which was the original total amount of chocolate blocks before we shared them in a ratio.

We have introduced ratio by sharing this amount physically, moving 4 blocks and 3 blocks at a time.

When teaching ratio for the first time, it is recommended that you use a physical set of items that can be shared by moving the parts, such as chocolate or counters.

Above there are some sharing in a ratio worksheets, which you can cut into parts using scissors.

Now try our lesson on Sharing in a Ratio using the Bar Model where we learn how to share an amount in a ratio using the bar model.

ratio bar model

Hundreds, Tens and Units

Place Value HTU Summary

We can count these counters more easily by collecting them into groups of hundreds, groups of ten and groups of units.

‘Unit’ means ‘one’ and each counter is one unit.

Each
rowA line going from left to right. A horizontal line.
contains a group of ten units.

We can count ten rows of ten units to make a total square of one hundred counters.

We only have one group of one hundred counters and therefore we write a ‘1’ in the hundreds column of our place value chart.

There are two rows of ten remaining so we can write a ‘2’ in the tens column of our place value chart.

We have six counters remaining so we write a ‘6’ in the units (or ones) column.

The total number is 126.

We form our number from the
digitsAny of the numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 that we use to make up our numbers
.

It is easier to count large numbers by counting the groups of hundreds, groups of tens and groups of ones.

Place Value HTU Example

We can count the total number of counters here by counting the groups of hundreds, tens and units.

We have 4 groups of hundreds.

We have 6 groups of ten.

We finally have 3 units (or ones).

We write our total by combining these
digitsAny of the numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 that we use to make up our numbers
: 4, 6 and 3 in order.

The total number of counters is 463.

Supporting Lessons

Introducing Tens and Units

Partitioning Hundreds, Tens and Units: Interactive Questions

Interactive Partitioning Cards for Hundreds, Tens and Units

Base Ten Blocks: Hundreds, Tens and Units

Interactive Base Ten Block Manipulatives

Representing Hundreds, Tens and Units with Base Ten Blocks

Hundreds, Tens and Units Worksheets and Answers

Hundreds tens and units worksheet answers

Blank Hundreds, Tens and Units (Ones) Place Value Charts

blank Hundreds tens and units place value chart for printing

blank Hundreds tens and ones place value chart for printing