Category: Uncategorized

Multiplying Fractions by simplifying first example

Before multiplying these fractions we can simplify the calculation by cancelling.

We look for a number that divides exactly into both a number on the top and a number on the bottom.

We can see that both 2 and 16 are even numbers and can be divided by 2.

We can halve 2 to get 1 and halve 16 to get 8.

Because we have halved a number on the top and halved a number on the bottom, the answer will be the same.

² / ₇ × ¹ / ₁₆ is the same as ¹ / ₇ × ¹ / ₈ .

Now that we have simplified the fractions we do the multiplication.

To multiply fractions we multiply the
numeratorsThe number on the top of a fraction, above the line.
together and then multiply the
denominatorsThe number on the bottom of a fraction, below the line.
together separately.

1 × 1 = 1 and 7 × 8 = 56.

¹ / ₇ × ¹ / ₈ = ¹ / ₅₆ .

Look for a number on the top and a number on the bottom that can both be divided by the same number.

Divide them both by this number before multiplying the fractions.

Multiplying Fractions example on cross cancelling

We look for a number on the top and number on the bottom that can both be divided by the same number.

5 and 10 are both in the 5 times table so can be divided by 5.

5 ÷ 5 = 1 and 10 ÷ 5 = 2.

We can also see that 9 and 12 are both in the 3 times table and can be divided by 3.

9 ÷ 3 = 3 and 12 ÷ 3 = 4.

⁵ / ₁₂ × ⁹ / ₁₀ is the same as ¹ / ₄ × ³ / ₂ .

Multiplying the numerators, 1 × 3 = 3.

Multiplying the denominators, 4 × 2 = 8.

⁵ / ₁₂ × ⁹ / ₁₀ = ³ / ₈ .

Supporting Lessons

Multiplying Fractions by Simplifying First: Interactive Questions

Multiplying Fractions by Simplifying First Worksheets and Answers

$Multiplying fractions by simplifying first worksheet pdf$

$multiplying fractions by simplifying first worksheet answers pdf$

Converting Fractions to Percentages

How to Multiply Fractions by Simplifying First

Multiplying Fractions by Simplifying First

To multiply fractions by simplifying first, use the following steps.

Find a number on top of the fractions and a number on the bottom of the fractions that can both be divided by the same number.

Divide both numbers in the fractions by this number.

Write the answers to this division in the place of the original numbers in the fraction.

Multiply the fractions like usual by multiplying the numerators and denominators separately.

Multiplying the numerators on top of the fractions equals the numerator on the top of the answer.

Multiplying the denominators on the bottom of the fractions equals the denominator on the bottom of the answer.

Here is an example of simplifying fractions before multiplying.

Here we have ⁴ / ₁₅ × ⁵ / ₉ .

The first step is to look for a number on the top of either of the fractions and a number on the bottom of either of the fractions that can both be divided by the same number.

We can see that both 5 on the top of the right fraction and 15 on the bottom of the left fraction are both in the 5 times table.

We say that 5 is a factor of both 5 and 15. A factor is a number that divides exactly into another number.

So we divide both 5 and 15 by 5.

$Example of multiplying fractions by simplifying first$

5 ÷ 5 = 1 and 15 ÷ 5 = 3.

We cross out the 5 and 15 and replace them with a 1 and a 3.

Now that we have simplified the fractions we can multiply them.

To multiply fractions, simply multiply the numerators and denominators separately.

The numerators are the numbers on the top.

4 × 1 = 4

The denominators are the numbers on the bottom.

3 × 9 = 27

⁴ / ₁₅ × ⁵ / ₉ = ⁴ / ₂₇ .

We know that the answer is fully simplified because no numbers divide into both 4 and 27.

Dividing the top and bottom of a fraction by the same number can be called cancelling the fraction.

When we cross off a number on the top of one fraction and the number on the bottom of another fraction diagonally, we can call it cross cancelling.

Why does Cross Cancelling Work?

We multiply by numbers on the top of fractions and divide by numbers on the bottom of fractions. This is because the line in a fraction means to divide by the number below it.

Cross cancelling divides the number we are multiplying by and the number we are dividing by by the same amount. This means that the answer does not change in size. The values have been multiplied by less but also divided by less so the answer remains the same as it would have before cross cancelling.

For example, here we have the multiplication of ² / ₇ × ¹ / ₁₆ .

$mutliplying fractions two sevenths times one sixteenth$

We can multiply the fractions without simplifying first.

We multiply the numerators on top first.

2 × 1 = 2

And multiply the denominators on the bottom.

7 × 16 = 112

Therefore ² / ₇ × ¹ / ₁₆ = ² / ₁₁₂ .

We can simplify the fraction afterwards by dividing both the numerator on top and the denominator on the bottom by 2.

² / ₁₁₂ = ¹ / ₅₆ , which is our final answer.

We simplified our answer by dividing the top and bottom by 2, just at the end of our process.

Instead we can simplify the fraction first by cross cancelling.

$mutliplying fractions two sevenths times one sixteenth looking at common factors$

The number 2 and the number 16 both can be divided by 2. This time, we will divide the top and the bottom by 2 before we multiply. 2 is a common factor of 2 and 16.

2 ÷ 2 = 1 and 16 ÷ 2 = 8.

$mutliplying fractions two sevenths times one sixteenth by dividing common factors by 2$

Multiplying the fractions ¹ / ₇ × ¹ / ₈ = ¹ / ₅₆ .

This gives us the same answer as before.

$multiplying fractions by cancelling common factors example$

We can also think of fractions as a multiplication and division sentence. We multiply by the numerators on top and divide by the denominators on the bottom.

² / ₇ × ¹ / ₁₆ is the same as 2 × 1 ÷ 7 ÷ 16.

¹ / ₇ × ¹ / ₈ is the same as 1 × 1 ÷ 7 ÷ 8.

We can compare the underlined digits to see that we have halved the number we are multiplying by from 2 to 1 but also halved the number we are dividing by from 16 to 8. The answer is the same.

Multiplying Fractions example of cross cancelling to multiply

Why do we Simplify Fractions before Multiplying Them?

Multiplying some fractions can result in large numbers being multiplied together. It is best to simplify fractions first by cross cancelling common factors to make the numbers smaller. Smaller numbers are easier to multiply.

You are also less likely to make a mistake by simplifying the fractions first. Even the final step of simplifying is easier because the numbers are smaller. Multiplying the fractions without simplifying first may result in very large numbers and it may not be obvious what to divide them by to simplify the fraction.

For example, here is ⁵ / ₁₂ × ⁹ / ₁₀ .

5 × 9 = 45 and 12 × 10 = 120

⁵ / ₁₂ × ⁹ / ₁₀ = ⁴⁵ / ₁₂₀ .

The numbers in the fraction were not particularly easy to multiply but the final step of simplifying the fraction is not easy as it is hard to see the highest common factor that divides into both 45 and 120.

In fact, both 45 and 120 can be divided by 15.

45 ÷ 15 = 3 and 120 ÷ 15 = 8.

However it is much easier to simplify the fractions before multiplying.

$Why we simplify fractions by cross cancelling first$

We can see that both 5 and 10 can be divided by 5 and 9 and 12 can be divided by 3.

⁵ / ₁₂ × ⁹ / ₁₀ = ¹ / ₄ × ³ / ₂ .

This results in much easier numbers to multiply and there is then no need to simplify a large fraction as the end.

¹ / ₄ × ³ / ₂ = ³ / ₈ .

Here is another example of why we simplify fractions before multiplying them.

We have ⁹ / ₁₄ × ⁴ / ₁₅ .

Again multiplying the fraction immediately can result in larger numbers.

9 × 4 = 36 and 14 × 15 = 210

⁹ / ₁₄ × ⁴ / ₁₅ = ³⁶ / ₂₁₀ which is not so easy to simplify.

We can divide by the common factor of 6 in 36 and 210.

36 ÷ 6 = 6 and 210 ÷ 6 = 35.

However it is much easier to simplify the fraction first to keep the numbers within the usual times tables.

Multiplying Fractions by simplifying them first

We can divide both 9 and 15 by 3 to get 3 and 5 respectively.

We can also divide 4 and 14 by 2 to get 2 and 7 respectively.

³ / ₇ × ² / ₅ = ⁶ / ₃₅ .

Because we already fully simplified the fractions first before multiplying them, there is no need to simplify our answer again. It is already in its simplest form.

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

$converting fractions to percentages$

Multiplying Proper Fractions

$why do fractions get smaller when we multiply them$

Finding ¹ / ₂ of ¹ / ₃ is the same as finding ¹ / ₂ × ¹ / ₃ .

¹ / ₃ means to divide the shape into 3 equal parts.

We now find one half of this by dividing it into two equal parts.

We now have 1 part out of 6 equal parts.

¹ / ₂ × ¹ / ₃ = ¹ / ₆ .

We can simply multiply the
numeratorsThe number on the top of a fraction, above the line.
of the fractions to get the numerator of the answer.

We can simply multiply the
denominatorsThe number on the bottom of a fraction, below the line.
of the fractions to get the denominator of the answer.

1 × 1 = 1, which goes on top.

2 × 3 = 6, which goes on the bottom.

Multiply the numerators on the top to get the numerator of the answer.

Multiply the denominators on the bottom to get the denominator of the answer.

Multiplying Fractions example of one half times one third

We have ¹ / ₂ × ¹ / ₃ .

Multiply the numbers on the tops of the fractions (the numerators) together.

1 × 1 = 1 and so, 1 is the numerator on the top of the answer.

Multiply the numbers on the bottoms of the fractions (the denominators) together.

2 × 3 = 6 and so, 6 is the denominator on the bottom of the answer.

¹ / ₂ × ¹ / ₃ = ¹ / ₆ .

Supporting Lessons

Multiplying Proper Fractions: Interactive Questions

Multiplying Proper Fractions Worksheets and Answers

$multiplying proper fractions worksheet pdf$

$multiplying proper fractions worksheet answers pdf$

Simplifying Fractions before Multiplying

How to Multiply Proper Fractions

A proper fraction is a fraction that has a smaller number on top (as the numerator) and larger number on the bottom (as the denominator).

To multiply proper fractions use the following steps:

Multiply the numerators on the top of the fractions.

This result will be the numerator on the top of the answer.

Multiply the denominators on the bottom of the fractions.

This result will be the denominator on the bottom of the answer.

Here is an example of multiplying the fractions of one half by one third.

$multiplying proper fractions one half times one third$

We are asked, “What is ¹ / ₂ × ¹ / ₃ ?”

The numerators are the numbers on the tops of the fractions. We multiply the numerator of the first fraction by the numerator of the second fraction to get the numerator on the top of our answer.

1 × 1 = 1

The denominators are the numbers on the bottoms of the fractions. We multiply the denominator of the first fraction by the denominator of the second fraction to get the denominator on the bottom of our answer.

2 × 3 = 6

$multiplying two proper fractions one half times one third equals one sixth$

¹ / ₂ × ¹ / ₃ = ¹ / ₆ .

$example of multiplying fractions one half times one third$

Here is another example of multiplying proper fractions using the steps above.

We have ¹ / ₄ × ¹ / ₃ .

First, we multiply the numerators.

1 × 1 = 1

So 1 is the numerator on top of our answer.

Next we multiply the denominators.

4 × 3 = 12

$Example of multiplying proper fractions one quarter times one third equals one twelfth$

¹ / ₄ × ¹ / ₃ = ¹ / ₁₂ .

In this example of multiplying proper fractions we have ² / ₅ × ² / ₃ .

We start by multiplying the numerators on the tops of the two fractions.

2 × 2 = 4

4 is the numerator on top of our answer.

Next we multiply the two denominators on the bottoms of the fractions together.

5 × 3 = 15

15 is the denominator on the bottom of our answer.

Multiplying Fractions example shown in steps

² / ₅ × ² / ₃ = ⁴ / ₁₅ .

In this example we are multiplying the proper fractions of ¹ / ₅ × ³ / ₄ .

Multiplying the numerators we have:

1 × 3 = 3

Multiplying the denominators we have:

5 × 4 = 20

Multiplying Fractions question

¹ / ₅ × ³ / ₄ = ³ / ₂₀ .

Why do Proper Fractions get Smaller when you Multiply them?

Multiplying a fraction by a proper fraction always makes it smaller.

This is because a proper fraction has a smaller number on top as its numerator compared to a larger number on the bottom as its denominator. When we multiply by a fraction we are multiplying by the number on top but dividing by the number on the bottom.

So we are multiplying by a smaller number than we are dividing by. This means that our answer will be smaller than we started with.

In this example we will start with the fraction ¹ / ₃ .

$the proper fraction one third$

We will find ¹ / ₂ of ¹ / ₃ .

This means that we can draw a line through the fraction of ¹ / ₃ , splitting it in half.

$half of the fraction one third$

We now have 6 parts in total rather than 3.

We have 1 part shaded out of 6 parts in total.

$multiplying proper fractions one half x one third is one sixth$

We can see that multiplying one third by one half made it smaller because it split the fraction in half.

We found a fraction of a fraction.

In Maths the word ‘of’ can be replaced with a multiplication.

Now try our lesson on Simplifying Fractions before Multiplying where we learn how to multiply larger fractions by simplifying them first.

$simplifying fractions before multiplying$

Multiplying by 9 and 19: Random Question Generator

Multiplying by 9 and 19: Question Generator – Maths with Mum Return to video lesson on Multiplying by 9 and 19

Multiplying by 4 and 8: Random Question Generator

Multiplying by 4 and 8: Question Generator – Maths with Mum Return to video lesson on Multiplying by 4 and 8

Times Tables Game (Hard)

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Time remaining = .

Your Score =

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Lesson: Times Tables Game (Extreme)

Multiplication Arrays

multiplication arrays showing the 2 times table

A multiplication array is a collection of items arranged in rows and columns that represent a number formed from a multiplication of two numbers.

The number represented is the total number of items in the rectangular array.

The multiplication represented is the number of items in each row multiplied by the number of items in each column.

There are 2 items in each column so each array here represents a multiple of 2.

There is 1 lot of 2 in the first array, which represents the multiplication of 1 × 2 = 2.

There are 2 lots of 2 in the second array, which represents the multiplication of 2 × 2 = 4.

There are 3 lots of 2 in the third array, which represents the multiplication of 3 × 2 = 6.

There are 4 lots of 2 in the fourth array, which represents the multiplication of 4 × 2 = 8.

Arrays can be used to teach multiplication visually and appreciate the size of the numbers formed in a multiplication.

Arrays can be used to show how multiplication is the same as repeated addition. In this case, we can see that we add a row of 2 each time.

Multiplication arrays can be used to provide students with a strategy to work out multiplications that they do not know.

Arrays can also be used to show how the order in which you multiply two numbers does not matter.

Multiplication arrays are a rectangular collection of objects arranged in rows and columns, which are used to represent a multiplication.

example of using multiplication arrays to represent 4 x 5 = 20

Here is a multiplication array used to represent the multiplication sentence of 4 x 5 = 20.

There are 4 counters in each column and there are 5 counters in each row.

We can see that we have taken the 4 counters in each column and repeated them 5 times.

We have multiplied 4 by 5.

There are 20 counters in total, so 4 x 5 = 20.

The number of counters in each column multiplied by the number of counters in each row must equal the total number of counters.

Supporting Lessons

Multiplication Arrays Worksheets and Answers

Multiplication as Equal Groups

Multiplication Arrays

What are Multiplication Arrays?

Multiplication arrays are rectangular collections of objects that are used to represent multiplication equations. The number of objects in each row is multiplied by the number of objects in each column to make the total number of objects in the array.

For example, here is the multiplication array showing 4 × 3.

a multiplication array showing the multiplication of 4 and 3

In a multiplication array, the number of items in each row multiplied by the total number of items in each column gives the total number of items.

There are 12 counters in total. The total number of items shown is 12.

There are 4 counters in each row and 3 counters in each column.

The multiplication array shows 4 lots of 3. 3 is mutliplied by 4 to make 12. The multiplication array represents 4 × 3 = 12.

It does not matter in which order we multiply the rows and columns.

Multiplication arrays can be used to show the commutative property of multiplication. The commutative property of multiplication means that it does not matter in which order the numbers are multiplied, the answer is still the same. For example 4 × 3 is the same as 3 × 4.

Here is the same array but grouped to show 3 × 4 instead of 4 × 3.

multipication array showing 3 × 4

This time, we can see that we have 3 lots of 4, whereas in the previous animation we have 4 lots of 3.

The result is still the same as the total number of counters did not change. Both 3 × 4 and 4 × 3 = 12.

Why do we Use Multiplication Arrays?

Multiplication arrays are used to introduce the concept of multiplication in a visual way. They help us to visualise the size of numbers produced in multiplication. Multiplication arrays also help us to understand multiplication as repeated addition and that the order in which two numbers are multiplied does not affect the size of the answer.

When teaching multiplication arrays, they can be introduced through the process of repeated addition.

For example, we can start with 2 counters, which represents 1 lot of 2. As we say “1 lot of 2” we can write 1 × 2.

We can then add another lot of two directly next to this to represent 2 lots of 2. As we say “2 lots of 2” we can write 2 × 2.

multiplication arrays to teach multiplication as repeated addition

The same applies for 3 lots of 2, written as 3 × 2 and 4 lots of 2, written as 4 × 2.

When teaching multiplication arrays, it helps to use physical items such as counters. You can also show examples of multiplication arrays in real life.

Some examples of multiplication arrays in real life are:

The number of eggs in an egg box.

The number of seats arranged in rows in a classroom.

The number of cakes placed on a baking tray.

It helps to show examples of arrays in real life to help learn multiplication facts.

Once you have introduced multiplication arrays as repeated addition, you should draw the link between the number of items in each row and each column to form the multiplication sentence.

multiplication arrays showing the two times table

Show that the number in each row multiplied by the number in each column is the number of counters we have in total.

This means that when you see multiplication arrays in real life, you can practice working out how many items there are in total by multiplying the number in each row by the number in each column.

Here is a similar example with the start of the 3 times table.

the 3 times table shown using multiplication arrays

1 lot of 3 is 3 and so, we write 1 × 3 = 3.

2 lots of 3 is 6 and so, we write 2 × 3 = 6.

3 lots of 3 is 9 and so, we write 3 × 3 = 9.

4 lots of 3 is 12 and so, we write 4 × 3 = 12.

learning multiplication using multiplication arrays

Multiplication arrays can also be used to teach the commutative property of multiplication. The commutative property of multiplication simply means that the order in which two numbers are multiplied does not affect the size of the answer. For example,

a multiplication array shown by 6 × 10

Here we have 6 multiplied by 10. We can see that we have 10 lots of 6 counters. There are 60 counters in total.

a multiplication array example

Here we have 6 lots of 10.

The result is still the same as the total number of counters did not change.

Now try our lesson on Multiplication as Equal Groups where we learn how multiplication can be used to represent equal groups of a number.

multiplication as equal groups

Multiplying by Multiples of 10 and 100

Multiplying by a Multiple of 10

multiplying by a multiple of 10 example of 5 × 20

A multiple of 10 is a number in the 10 times table.

To multiply a whole number by 10, simply write a zero digit on the end of it.

To multiply a number by a multiple of 10, multiply the number digits in front of the 0 digit and then write a 0 on the end.

20 is a multiple of 10. 20 is 2 × 10.

To work out 5 × 20, we can work out 5 × 2 × 10.

5 × 2 = 10 and now we just need to multiply by 10.

To multiply 10 by 10, we simply write a 0 digit on the end.

10 × 10 = 100 and so, 5 × 20 = 100.

Multiplying by a Multiple of 100

multiplying by multiples of 100 example of 7 × 400

A multiple of 100 is a number in the 100 times table.

To multiply a whole number by 100, simply write two 0 digits on the end of it.

To multiply a number by a multiple of 100, multiply the number by the digits in front of the two 0 digits and then write two 0 digits on the end.

400 is a multiple of 100. 400 = 4 × 100.

To work out 7 × 400, we can work out 7 × 4 × 100.

7 × 4 = 28 and now we just need to multiply by 100.

To multiply 28 by 100 we simply write two 0 digits on the end.

28 × 100 = 2800 and so, 7 × 400 = 2800.

Multiply the number by the first digits of the multiple of 10 or 100.

Write the same number of zeros on the end of the answer as there are in 10 or 100.

multiplying by multiples of 100 example of 6 × 110 = 660

110 is a multiple of 10 because it ends in one 0 digit.

110 = 11 × 10 so to multiply 6 by 110, we will multiply 6 × 11 × 10.

6 × 11 = 66 and we will multiply by 10 by writing a 0 on the end.

66 × 10 = 660 and so, 6 × 110 = 660.

Supporting Lessons

Multiplying by Multiples of 10 and 100: Interactive Questions

Multiplying by Multiples of 10 and 100 Worksheets and Answers

multiplying by multiples of 10 and 100 worksheet pdf

Multiplying by 4 and 8 Using Doubling

Multiplying by Multiples of 10 and 100

What is a Multiple of 10?

A multiple of 10 is a number in the 10 times table. Multiples of 10 will always have a last digit of 0. The first few multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100.

We can recognise multiples of 10 because they are numbers that end in at least one 0 digit.

What is a Multiple of 100?

A multiple of 100 is a number in the 100 times table. Multiples of 100 will always have two 0 digits as their last two digits. The first few multiples of 100 are 100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000.

We can recognise multiples of 100 because they are numbers that end in at least two 0 digits.

How to Multiply by Multiples of 10

To multiply a whole number by a multiple of 10, follow these steps:

Multiply the whole number by the digits in front of the last 0 in the multiple of 10.

Multiply this result by 10 by writing a 0 digit on the end.

For example, we will multiply 5 × 20.

5 x 20

20 is a multiple of 10 because it ends in a 0 digit.

We can write multiples of 10 as the digits in front of the zero digit multiplied by 10. For example, 20 = 2 × 10.

multiplying by 20 is the same as multiplying by 2 and then by 10

Multiplying by 20 is the same as multiplying by 2 and then by 10.

To multiply 5 × 20, we multiply 5 × 2 × 10.

5 x 2 x 10

5 × 2 = 10.

10 x 10

10 × 10 = 100. To multiply a whole number by 10, simply write a 0 digit on the end. 10 becomes 100.

10 x 10 = 100

5 × 20 = 5 × 2 × 10 = 10 × 10 = 100. Therefore 5 × 20 = 100.

To multiply by a multiple of 10, simply multiply the digits ignoring the last 0 digit and then write the 0 digit back in at the end.

Because 5 × 2 = 10, 5 × 20 = 100.

example of multiplying by a multiple of 10, 5 × 20

Here is another example of multiplying by a multiple of 10. We have 6 x 110.

110 is a multiple of 10 because it ends in a 0.

example of how to multiply a number by a multiple of 10, 6 x 110 = 660

6 x 11 = 66 and so, 6 x 110 = 660.

How to Multiply by Multiples of 100

To multiply a whole number by a multiple of 100, follow these steps:

Multiply the whole number by the digits in front of the last two 0 digits in the multiple of 100.

Multiply this result by 100 by writing two 0 digits on the end.

For example, here is 7 × 400. 400 is a multiple of 100 because it ends in 2 zero digits.

7 x 400

We multiply 7 by 4 and then write two 0 digits on the end to multiply by 100.

Multiplying by 400 is the same as multiplying by 4 and then 100.

multiplying by 400 is the same as multiplying by 4 and then by 100 width=

7 x 400 is the same as 7 x 4 x 100.

4 x 7 x 100

4 x7 = 28. We now need to multiply by 100.

To multiply a whole number by 100, simply write two 0 digits on the end.

28 x 100 = 2800

28 x 100 = 2800. Therefore 7 x 400 = 2800.

multiplying by multiples of 100 example of 7 x 400

Because 7 x 4 = 28, 7 x 400 = 2800.

Here is another example of multiplying by a multiple of 100. We have 12 x 500.

500 is a multiple of 100 because it ends in 2 zero digits.

example of multiplying by a multiple of 100, 12 x 500 = 6000

12 x 5 = 60 and so, 12 x 500 = 6000.

We simply write 2 more zeros on the end of 60 to make 6000.

Now try our lesson on Multiplying by 4 and 8 Using Doubling where we learn how to use doubling to multiply by 4 and 8.

multiplying by 8 by doubling doubling and doubling again

How to Find Isosceles Triangle Angles

Angles in an Isosceles Triangle

Angles Isosceles Triangle equal sides equal angles diagram

An isosceles triangle is a type of triangle that has two sides that are the same length.

The two marked sides are both the same length.

The two angles opposite these two marked sides are also the same: both angles are 70°.

All three interior angles add to 180° because it is a triangle.

An isosceles triangle has two equal sides and angles.

These two equal sides will be marked with short lines.

Missing angles in an isosceles triangle example

The triangle above is isosceles because there are lines marking two of its equal sides.

Angle ‘a’ and the angle marked 50° are opposite the two equal sides.

Angle ‘a’ must be 50° as well.

The two equal angles, 50° and 50°, add to make 100°.

To find angle ‘b’, we subtract 100° from 180°. This equals 80°.

Supporting Lessons

Angles in an Isosceles Triangle Worksheets and Answers

angles in an isosceles triangle worksheet pdf

Forming Algebraic Expressions

Angles in an Isosceles Triangle

What is an Isosceles Triangle?

An isosceles triangle is a triangle that has two equal sides and two equal angles. The two equal sides are marked with lines and the two equal angles are opposite these sides.

We can recognise an isosceles triangle because it will have two sides marked with lines.

Below is an example of an isosceles triangle.

It has two equal sides marked with a small blue line.

It has two equal angles marked in red.

An isosceles triangle with its equal sides and base angles marked

We can see that in this above isosceles triangle, the two base angles are the same size.

All isosceles triangles have a line of symmetry in between their two equal sides.

The sides that are the same length are each marked with a short line.

The two equal angles are opposite to the two equal sides.

The angle at which these two marked sides meet is the odd one out and therefore is different to the other two angles.

An isosceles triangle that has one base angle 70 degrees has the other base angle as 70 degrees

If we are told that one of these marked angles is 70°, then the other marked angle must also be 70°.

How to Find a Missing Angle in an Isosceles Triangle

To find a missing angle in an isosceles triangle use the following steps:

If the missing angle is opposite a marked side, then the missing angle is the same as the angle that is opposite the other marked side.

If the missing angle is not opposite a marked side, then add the two angles opposite the marked sides together and subtract this result from 180.

This is because all three angles in an isosceles triangle must add to 180°

For example, in the isosceles triangle below, we need to find the missing angle at the top of the triangle.

finding a missing angle at the top of an isosceles triangle example of 40 degrees

The two base angles are opposite the marked lines and so, they are equal to each other.

Both base angles are 70 degrees.

The missing angle is not opposite the two marked sides and so, we add the two base angles together and then subtract this result from 180 to get our answer.

70° + 70° = 140°

The two base angles add to make 140°.

Angles in an isosceles triangle add to 180°.

We subtract the 140° from 180° to see what the size of the remaining angle is.

180° – 140° = 40°

Finding the missing top angle of an isosceles triangle to be 40 degrees by subtracting the two equal base angles which are 70 degrees

The missing angle on the top of this isosceles triangle is 40°.

We can also think, “What angle do we need to add to 70° and 70° to make 180°?”

The answer is 40°.

How to Find Missing Angles in an Isosceles Triangle from only One Angle

If only one angle is known in an isosceles triangle, then we can find the other two missing angles using the following steps:

If the known angle is opposite a marked side, then the angle opposite the other marked side is the same. Add these two angles together and subtract the answer from 180° to find the remaining third angle.

If the known angle is not opposite a marked side, then subtract this angle from 180° and divide the result by two to get the size of both missing angles.

Here is an example of finding two missing angles in an isosceles triangle from just one known angle.

We know that one angle is 50°. This angle is opposite one of the marked sides.

This means that it is the same size as the angle that is opposite the other marked side. This is angle ‘a’.

Therefore angle ‘a’ is 50° too.

Finding the missing angles in an isosceles triangle when we know one angle example

Now to find angle ‘b’, we use the fact that all three angles add up to 180°.

To find angle ‘b’, we subtract both 50° angles from 180°. We first add the two 50° angles together.

50° + 50° = 100°

and 180° – 100° = 80°

Angle ‘b’ is 80° because all angles in a triangle add up to 180°.

Here is another example of finding the missing angles in isosceles triangles when one angle is known.

This time, we know the angle that is not opposite a marked side. We have 30°.

An isosceles triangle on its side with top angle being 30 degrees

We can subtract 30° from 180° to see what angle ‘a’ and ‘b’ add up to.

180° – 30° = 150°

And so, angles ‘a’ and ‘b’ both add up to 150°.

Subtracting the top angle 30 degrees from an isosceles triangle to leave the two base angles

Because angles ‘a’ and ‘b’ are both opposite the marked sides, they are equal to each other.

The size of these two angles are the same.

We divide 150° into two equal parts to see what angle ‘a’ and ‘b’ are equal to.

150° ÷ 2 = 75°

This is because 75° + 75° = 150°.

Working out the two base angles of an isosceles triangle by subtracting the top angle and dividing by 2

Angles ‘a’ and ‘b’ are both 75°.

finding the missing angles in an isosceles triangle example where one angle is known.

We can see that the three angles in an isosceles triangle add up to 180°.

75° + 75° + 30 = 180°.

Now try our lesson on Forming Algebraic Expressions where we learn how to write algebraic expressions.

algebraic epressions

Short Division with Decimal Remainders

how to do short division with decimal remainders example

how to write a remainder as a decimal

To perform schort division, divide each
digitAny of the numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 that can be used to make a number.
in the number by the digit written on the left.

We divide each digit in 846 by 4, working from to left to right.

8 ÷ 4 = 2. We write the 2 above the line.

Next, 4 ÷ 4 = 1 and we write this above the line.

6 ÷ 4 = 1 remainder 2. This is because we need 2 more from 4 to make 6.

This remainder is carried over to the tenths column.

Since there are no tenths in the original number of 846, we write a zero in the tenths column.

Now we can carry the remainder of 2 over to make ’20’ tenths, which can then be divided by 4.

20 ÷ 4 = 5 and so, we write 5 in the tenths column above the line.

846 ÷ 4 = 211.5.

Write a zero after the decimal point and carry the remainder over for division.

example of a short division with a decimal remainder

4 ÷ 4 = 1.

8 ÷ 4 = 2.

7 ÷ 4 = 1 remainder 3.

Since we have a remainder, we need to write a zero in the first decimal column and carry the remainder of 3 over to make 30 tenths.

30 ÷ 4 = 7 remainder 2.

Since we still have a remainder, we write another zero in the next column along and carry the 2 over to make 20 hundredths

20 ÷ 4 = 5. Since there are no more remainders, we have our final answer.

487 ÷ 4 = 121.75.

Supporting Lessons

Short Division with Decimal Remainders: Interactive Questions

Short Division with Decimal Remainders Worksheets and Answers

short division with decimal remainders worksheet pdf