What do the Angles in a Quadrilateral Add Up to?

The four angles in a quadrilateral always add up to 360°. This rule is true for all quadrilaterals.

A quadrilateral is any 4-sided shape. Quadrilaterals also always have 4 angles.

For example, the 4 angles in a rectangle are all 90°. Four lots of 90° is 360°.

Here are some more examples of quadrilaterals where the angles are shown adding up to 360°.

How to Find a Missing Angle in a Quadrilateral

The 4 angles in a quadrilateral always add up to 360°. To find a missing angle in a quadrilateral, add up the 3 known angles and subtract this result from 360°.

For example, here is a quadrilateral containing 3 known angles of 75°, 85° and 140°.

The first step is to add up the 3 known angles. 75° + 85° + 140° = 300°.

The second step is to subtract this result from 360°.

360° – 300° = 60° and so, the missing angle is 60°.

How to Find Angles in a Parallelogram

If one angle in a parallelogram is known, all other angles can be calculated. To find missing angles in a parallelogram, use the following rules:

• Opposite angles are equal.
• Adjacent angles add to 180°.

In the example below, one angle is known to be 120°.

The angle opposite to this angle is the same size and so, angle a = 120°.

Angles b and c are both next to the 120° angle.

Therefore 120° + b = 180° and angle b = 60°.

120° + c = 180° and angle c = 60°.

b and c are opposite to each other, so we can see that they too are equal.

We can check the angles in our parallelogram by making sure that all 4 angles add to 360°.

120° + 120° + 60° + 60° = 360° and so, the result is correct.

Sum of Angles in a Quadrilateral Formula

The formula for the sum of the angles in a polygon is (n-2) × 180°, where n is the number of sides. A quadrilateral has 4 sides and so, n = 4. The formula (n-2) × 180° becomes 2 × 180° = 360°. Therefore the sum of the angles in a quadrilateral is 360°.

The formula for the sum of the angles in any polygon is (n-2) × 180°.

The (n-2) part of the formula tells us how many triangles can be drawn inside any polygon.

n is the number of sides the shape has. Every time a new side is added, a new triangle can be formed.

In a quadrilateral, we can only form 2 triangles. For a quadrilateral, n = 4 and so the (n-2) becomes 4 – 2 which equals 2.

We have 2 triangles, so 2 lots of 180°.

2 × 180° = 360° and so, all 4 angles in a quadrilateral add to 360°.

Why do Angles in a Quadrilateral Sum to 360 Degrees?

Angles in a quadrilateral add to 360° because two triangles can be made inside any quadrilateral by drawing straight lines from one corner to the other corners. Each triangle contains 180° and so, two triangles contain 360°.

In the example below, we see the four-sided shape divided into two triangles.

The 3 angles in each triangle add up to 180°.

The combination of both triangle angles makes the full angles of the quadrilateral.

2 lots of 180° is 360°.

Therefore the angles in a quadrilateral sum to 360°.

All quadrilaterals sum to 360° because we can always draw two triangles inside them by connecting two opposite corners.

Exterior Angles of a Quadrilateral

The exterior angles of a quadrilateral always add up to 360°. If the quadrilateral is shrunk down to the size of a point, all that is left are the exterior angles around the outside of it. Angles around a point add to 360° and so, the exterior angles add to 360°.

In the image below, we can see the exterior angles of a quadrilateral marked.

We can see that these exterior angles encircle a point completely.

Angles around a point add to 360° because 360° is a full turn.

All 4 exterior angles on a quadrilateral add to 360°.

This rule is true for all polygons. The exterior angles of any polygon always add up to 360°.

What do Angles in a Pentagon Add Up To?

The sum of the 5 interior angles in any pentagon is always equal to 540°.

We can see that the angles in the pentagon below all add up to 540°.

How to Find a Missing Angle in a Pentagon

To find a missing angle in a pentagon, add up all of the other known angles then subtract this sum from 540°.

For example, here is a pentagon with known angles of 110°, 130°, 80° and 160°.

The first step is to add the known angles together.

110 + 130 + 80 + 160 = 480°.

The second step is to subtract this total from 540°.

540° – 480° = 60° and so, the missing angle is 60°.

Here is an example of finding more than one missing angle in a pentagon.

In this example, the angles in the pentagon have lines on them. This tells us that some angles are the same size.

The two angles that both have one line on them are equal to each other. Therefore angle a = 120°.

We also know that angles b and c are equal to each other because they both have 2 lines on them.

To work out the size of angles b and c, we will first find the sum of the three angles above.

100° + 120 ° + 120° = 340°.

We can subtract this from 540° to get 200°.

This means that angles b and c must add together to make 200°. Since they are both the same size, we will just divide 200° by 2.

Angle b = 100° and angle c = 100°.

Angles in a Regular Pentagon

Each interior angle in a regular pentagon is equal to 108°. This is because the sum of all 5 interior angles in any pentagon is 540°. In a regular pentagon, all five angles are of equal size and so, we divide 540° by 5 to get 108°.

Below is a regular pentagon.

If a shape is regular, this means that all of its sides are the same length and all of its angles are the same size.

Angles in a pentagon add up to 540°.

Since all 5 angles are the same size, the total of 540° is shared evenly into 5 equal parts.

540° ÷ 5 = 108° and so each angle is 108°.

We can check the result by adding our angles.

108° + 108° + 108° + 108° + 108° = 540°.

Formula to Find the Angles in a Pentagon

The formula for the sum of interior angles in a polygon is (n-2) × 180°, where n is the number of sides. A pentagon has 5 sides and so, n = 5. The formula (n-2) × 180° becomes 3 × 180° = 540°. Therefore the sum of angles in a pentagon is 540°.

The formula works because it tells us how many triangles can be formed inside each shape.

The number of triangles that can be drawn is equal to 2 less than the number of sides, or n – 2.

Each triangle contributes 180° to the sum of the interior angles and so, we have (n-2) × 180°.

Why do Angles in a Pentagon Add to 540?

Angles in a pentagon add to 540° because three triangles can be made inside any pentagon by drawing lines from one corner to each of the other corners. Each triangle contains 180° and 3 × 180° = 540°.

Below is a pentagon divided into 3 triangles.

These triangles have been formed by taking one corner of the shape and drawing straight lines to each of the other corners.

Each triangle contains 180°.

180° + 180° +180° = 540° and so, the sum of the three triangles is 540°.

The angles in all three triangles form the interior angles of the pentagon and so the sum of angles in a pentagon equals 540°.

Exterior Angles of a Pentagon

Exterior angles of all pentagons add up to 360°. In a regular pentagon, each exterior angle is 72°. This is because each angle is the same size and 360° ÷ 5 = 72°.

Exterior angles of all polygons always add up to 360°.

We can see the 5 exterior angles of a regular pentagon marked below.

Exterior angles add up to 360° because each polygon can be shrunk down until it forms a point. All that is left are the exterior angles. Angles around a point add to 360° and so, the exterior angles add to 360°.

This is shown below.

What are 2-Step Equations?

2-step equations are equations that can be solved in 2 steps. 2𝑥 + 2 = 10 is an example of a 2-step equation. The first step is to subtract 2 from both sides to get 2𝑥 = 8. The second step is to divide both sides by 2 to get 𝑥 = 4. The equation is solved because the variable is on its own with no number next to or in front of it.

Two steps are required to solve the equation, which means that 2 calculations are done to find the value of the variable.

In the 2-step equation of 2𝑥 + 2 = 10, we first subtract 2 and then divide by 2. This is 2 steps.

The solution to an equation is the number that the variable (letter) needs to be to make both sides of the equation equal to the same value.

In 2𝑥 + 2 = 10, this only occurs when 𝑥 = 4.

When 𝑥 = 4, we get 2 × 4 + 2 which equals 10. This means that both sides of the equation equal 10.

We can think of solving a 2-step equation as asking us, “What value does 𝑥 need to be to make both sides of the equation the same?”.

How to Solve 2-Step Equations

To solve 2-step equations, use these steps:

1. Add or subtract the same number on both sides of the equation so that there is one term remaining.
2. Multiply or divide both sides of the equation by the same number.
3. The equation is solved when the variable (letter) is alone on one side of the equals sign.

For example, in the 2-step equation of 6𝑥 – 5 = 7, we first look to remove the ‘-5’.

We always use the inverse operation to remove a term. The inverse means the opposite. The opposite of subtracting 5 is to add 5 and so, we add 5 to both sides of the equation.

On the left-hand side of the equation, 6𝑥 – 5 + 5 becomes 6𝑥. This is because the -5 and + 5 cancel each other out.

When solving equations it is important that whatever is done to one side of the equation is done to the other side of the equation as well. We must also add 5 to the right-hand side.

On the right-hand side of the equation, 7 + 5 = 12.

This leaves us with 6𝑥 = 12.

We now wish to remove the ‘6’ from in front of the 𝑥. 6𝑥 means that 𝑥 is multiplied by 6 and so, the inverse operation is to divide by 6. We must do this to both sides of the equation.

On the left-hand side of the equation, 6𝑥 ÷ 6 = 𝑥.

On the right-hand side of the equation, 12 ÷ 6 = 2.

We have 𝑥 = 2.

Because we have isolated 𝑥 on its own, we have found our solution. The solution to this 2-step equation is 𝑥 = 2.

How to Check 2-Step Equations

To check the answer to a 2-step equation, substitute it back into the equation in place of the variable. Work out the value of each side of the equation. If both sides of the equation equal the same number, the answer is correct. If they do not equal the same number, the answer is wrong.

For example, the solution to 6𝑥 – 5 = 7 is 𝑥 = 2.

To check if the solution is correct, substitute it back into the original equation in place of 𝑥.

6𝑥 means 6 × 𝑥 and so, 6𝑥 – 5 becomes 6 × 2 – 5 when we substitute the 𝑥 for a 2.

6 × 2 – 5 = 7, which is the same result as on the right-hand side of the equals sign and so, we know this solution is correct.

For instance, if instead we had made a mistake and found the solution to 6𝑥 – 5 = 7 as 𝑥 = 1, we would not get an answer of 7 when we put 𝑥 = 1 back into the equation.

6 × 1 – 5 = 1 and because this is not equal to 7, we would know that 𝑥 = 1 was an incorrect answer.

2-Step Equations with Fractions

^𝑥/₃ + 8 = 14 is an example of a 2-step equation involving fractions. The first step is to subtract 6 from both sides to leave ^𝑥/₃ = 6. The second step is to multiply both sides by the denominator of 3 to leave 𝑥 = 18.

To solve a 2-step equation with a fraction, we multiply both sides of the equation by the denominator of the fraction. This is because the fraction tells us to divide the numerator on top by the denominator on the bottom.

^𝑥/₃ means 𝑥 ÷ 3.

The first step is to remove the ‘+8’ by doing the inverse. The inverse of adding 8 is to subtract 8. We subtract 8 from both sides of the equation.

We are left with ^𝑥/₃ = 6.

^𝑥/₃ means 𝑥 ÷ 3. We do the inverse of dividing by 3, which is to multiply by 3. We do this to both sides of the equation.

We get 𝑥 = 18 as the solution to this 2-step equation with a fraction.

Here is another example of a 2-step equation involving a fraction. We have ^𝑥/₅ – 1 = 11

The first step for solving this equation is to remove the ‘-1’. The inverse of subtracting 1 is to add 1. We add 1 to both sides of the equation.

This leaves ^𝑥/₅ = 12.

^𝑥/₅ means 𝑥 ÷ 5. The inverse of dividing by 5 is to multiply by 5. We multiply both sides of the equation by 5.

This gives us the solution of 𝑥 = 60.

2-Step Equations with Brackets

3(𝑥 + 2) = 12 is an example of a 2-step equation involving brackets. The first step is to divide both sides of the equation by the number in front of the brackets to leave 𝑥 + 2 = 4. The second step is to subtract 2 from both sides to leave 𝑥 = 2.

The number in front of the brackets multiplies the brackets by this number. 3(𝑥 + 2) means 3 × (𝑥 + 2).

We want to remove the 3 from in front of the brackets. The opposite of multiplying by 3 is to divide by 3. We divide both sides of the equation by 3.

This leaves 𝑥 + 2 = 4.

We now remove the ‘+2’. The inverse of adding 2 is to subtract 2. We subtract 2 from both sides of the equation.

The solution is 𝑥 = 2.

Here is another example of a 2-step equation with brackets. We have 5(𝑥 – 7) = 15.

The first step is to remove the 5 from in front of the brackets. The brackets are being multiplied by 5 and the inverse of multiplying by 5 is to divide by 5. We divide both sides of the equation by 5.

This leaves 𝑥 – 7 = 3.

We now want to remove the ‘-7’. The inverse of subtracting 7 is to add 7. We add 7 to both sides of the equation.

The solution is 𝑥 = 10.

How to Divide Decimals by Decimals Step by Step

To divide decimals by decimals, follow these steps:

1. Look at the number after the division sign and move its decimal point to the right to make it a whole number.
2. Move the decimal point in the number before the division sign by the same number of places.
3. Divide these new numbers using short division.

For example, in 0.35 ÷ 0.007, the decimal point is moved 3 places right in both numbers to make the division become 350 ÷ 7.

350 ÷ 7 = 50 and so, 0.35 ÷ 0.007 also is equal to 50.

Dividing Decimals without Remainders

2.4 ÷ 0.03 is an example of dividing decimals where there are no remainders. 0.03 is multiplied by 100 to make it become 3 and 2.4 is multiplied by 100 to make it become 240. 240 ÷ 3 = 80 and so, 2.4 ÷ 0.03 = 80. This answer is exact and contains no remainders.

0.03 is multiplied by 100 to make it a whole number. This is because 100 has 2 zeros in it and there are 2 digits after the decimal point in 0.03.

We must also multiply 2.4 by 100 to keep the answer the same. 2.4 multiplied by 100 is 240.

We can now do the division. 240 ÷ 3 = 80 and so, 2.4 ÷ 0.03 is also 80.

Dividing Decimals with Remainders

0.561 ÷ 0.06 is an example of dividing decimals involving remainders. Both 0.06 and 0.561 are multiplied by 100 to make the division become 56.1 ÷ 6. We then divide digit by digit. 6 does not divide into 5 and so, the 5 is carried as a remainder to make 56. 56 ÷ 6 = 9, remainder 2. The 2 is carried over to the 1 to make 21. 21 ÷ 6 = 3, remainder 3. Another zero must be added and the remainder of 3 is carried over to make 30. 30 ÷ 6 = 5 and the division is complete. 0.561 ÷ 0.06 = 9.35.

0.06 is multiplied by 100 to make it a whole number of 6. This is because 0.06 contains 2 digits after the decimal point and 100 contains 2 zeros.

We also multiply 0.561 by 100 by moving the decimal point 2 places to the right, making 56.1.

56.1 ÷ 6 can be done using short division. We divide each digit in 56.1 by 6 individually.

Dividing Decimals Examples

2.0223 ÷ 0.003 is an example of dividing decimals. To make 0.003 a whole number, we multiply it by 1000 since 0.003 contains 3 digits after the decimal point and 1000 contains 3 zeros. We also multiply 2.0223 by 1000 by moving the decimal place 3 places right to become 2022.3. We then divide 2022.3 by 3 using short division to get an answer of 674.1.

2022.3 is divided using short division by dividing each digit individually by 3 from left to right.

2 ÷ 3 = 0, remainder 2. We carry the 2 over to the 0 to make 20.

20 ÷ 3 = 6, remainder 2. We carry the 2 over to the 2 to make 22.

22 ÷ 3 = 7, remainder 1. We carry the 1 over to the 2 to make 12.

12 ÷ 3 = 4. We write the decimal point after this digit, directly above the decimal point in the example.

Finally, 3 ÷ 3 = 1.

20223 ÷ 3 = 674.1 and so, 2.0223 ÷ 0.003 = 674.1.

How to Decide Whether to Measure Something in Grams or Kilograms?

1 kilogram weighs the same as 1 litre of water. If something weighs 1 kilogram or more, it makes sense to measure it in kilograms. If something is less than a kilogram, it makes sense to measure it in grams. A good rule to follow is if the object is too heavy to hold high in one hand, measure it in kilograms. If the object can be held easily in one hand, measure it in grams.

All objects can be measured in both grams or kilograms but it is best to choose a unit that will not be written as a very small decimal number or a number that is larger than 1000.

For example, a strawberry, a pencil and a sock are all objects that can be held easily in one hand and so, they should be measured in grams.

A bowling ball would be heavy to hold up high with one hand and so, kilograms would be a better unit of mass. Both a car and a man are impossible to hold up high with one hand and therefore, kilograms would be the best unit of mass to use.

It is better to say that a pencil is 10 grams than it is to say that it is 0.01 kilograms. In general, people prefer to use numbers that are not decimals, especially decimals that are less than 0.

It is better to say that a man weighs 80 kilograms than it is to say that a man weighs 80 000 grams. Numbers larger than 1000 are harder to comprehend and compare.

Here is a table showing some examples of objects measured in grams and kilograms.

How Heavy is a Gram?

One gram is very light. One gram is as heavy as a blueberry or a paperclip. A pencil weighs about 10 grams, an apple weighs about 100 grams and a loaf of bread weighs about 500 grams.

Each gram can be thought of as being the same weight as a paperclip. So if something weighs 10 grams, it weighs the same as 10 paperclips.

Grams are written for short as g. 1 gram is written as 1 g. 10 grams is written as 10 g.

A gram is the same the weight as a blueberry.

An acorn weighs about 5g.

This means that an acorn weighs about the same as 5 paperclips.

An apple weighs about 100 grams.

This means that an apple weighs the same as 100 paperclips or 20 acorns.

A loaf of bread weighs about 500 grams.

This means that a loaf of bread weighs the same as 500 paperclips or 5 apples.

How Heavy is One Kilogram?

One kilogram by definition is the weight of 1 litre of water. Some other objects that weigh 1 kilogram are a pineapple, a standard bag of sugar and a toaster. A large family drink bottle weighs 2 kilograms, a house cat weighs 5 kilograms and a large dog weighs about 30 kilograms.

A kilogram is 1000 grams. The suffix of kilo means 1000.

If one loaf of bread weighs 500 grams, then two loaves of bread weigh 1 kilogram. This is because 500 + 500 = 1000. 1000 grams is 1 kilogram.

An adult man weighs 80 kg and so, an adult man weighs about the same as 80 one-litre bottles of water.

A car weighs about 1500 kg and so, a car weighs about the same as 20 adult men or 1500 one-litre bottles of water.

How to Convert from Grams to Kilograms

To convert from grams to kilograms, divide the amount by 1000. This can be done by moving the decimal point 3 places to the left. For example, 375 g = 0.375 kg.

Here are some examples of converting from grams to kilograms by dividing by 1000.

How to Convert from Kilograms to Grams

There are 1000 grams in a kilogram, so to convert from kilograms to grams, simply multiply the amount by 1000. For example 5.2 kg = 5200 g.

Here are some examples of converting from kilograms to grams by multiplying by 1000.

How to Round to a Given Number of Decimal Places

To round to a given number of decimal places, use these steps:

1. Count the same number of digits after the decimal point as the number of decimal places being rounded to.
2. Draw a vertical line immediately after this digit.
3. If the digit to the right of this line is 5 or more, increase the digit to the left of the line by 1.
4. If the digit to the right of this line is 4 or less, keep the digit to the left of the line the same.
5. Remove all digits that come after the line.

For example, round 2.8397 to 3 decimal places.

The first step is to count 3 digits after the decimal point to find the 3rd decimal place. This is the digit of 9.

The second step is to draw a line after the 9.

The 3rd step is to look at the digit to the right of this line, which is a 7.

7 is ‘5 or more’ and so, it rounds this number up.

We increase the digit to the left of the line by 1. Since this digit is a 9, we look at the digit in front of this as well. We look at the digits of 3 and 9 in 2.8397 and increase 39 to 40.

We then remove all digits from after the line.

2.8397 rounds up to 2.840 when written to 3 decimal places.

How to Round to the Nearest Tenth

To round to the nearest tenth, look at the 2nd digit after the decimal point (the hundredths column). If this digit is 5 or more, increase the 1st digit after the decimal point (the tenths column) by 1. If the hundredths digit is 4 or less, keep the tenths digit the same. Finally, remove all digits after the tenths column.

Rounding to the nearest tenth means the same as rounding to 1 decimal place.

This is because the tenths column is the first digit after the decimal point.

For example, round 0.5814 to the nearest tenth. The choice is to round up to 0.6 or to round down to 0.5.

The first step is to look at the 2nd digit after the decimal point.

In this example, this is an 8.

The next step is to decide whether to round up or down depending on whether this digit is 5 or more or 4 or less.

8 is in the ‘5 or more’ category and so, we round up.

To round up, increase the tenths digit by 1 and remove the digits after the tenths column.

0.5814 rounds up to 0.6 when rounded to the nearest tenth.

Here are some examples of rounding to 1 decimal place (to the nearest tenth).

Notice that if the digit in the 2nd decimal place is a 5, we still round up. For example, 0.25 round up to 0.3.

How to Round to the Nearest Hundredth

To round to the nearest hundredth, look at the 3rd digit after the decimal point (the thousandths column). If this digit is 5 or more, increase the 2nd digit after the decimal point (the hundredths column) by 1. If the thousandths digit is 4 or less, keep the hundredths digit the same. Finally, remove all digits after the hundredths column.

Rounding to the nearest hundredth means the same as rounding to 2 decimal places.

For example, round 1.9021 to the nearest hundredth. The 2nd decimal place is 0 so the choice is to round up to 1.91 or down to 1.90.

The first step is to look at the digit in the 3rd decimal place, which is the 2.

The next step is to decide whether to round up or down depending on whether this digit is 5 or more or 4 or less.

2 is in the ‘4 or less’ category and so, rounds down.

To round down, keep the hundredths digit the same and remove the digits after this.

The hundredths digit is 0 and so, we keep this the same and remove the 2 and the 1 from after it.

Here are some examples of rounding to 2 decimal places.

How to Round to the Nearest Thousandth

To round to the nearest thousandth, look at the 4th digit after the decimal point (the ten-thousandths column). If this digit is 5 or more, increase the 3rd digit after the decimal point (the thousandths column) by 1. Instead, if this digit is 4 or less, keep the thousandths digit the same. Finally, remove all digits after the thousandths column.

Rounding to the nearest thousandth means the same as rounding to 3 decimal places.

For example, round 3.3685 to the nearest thousandth. The 3rd digit after the decimal point is 8 and so the choice is to round up to 3.369 or down to 3.368.

The first step is to look at the 4th digit after the decimal point, which is 5.

If this digit is 5 or more, round up and if it is 4 or less, round down.

5 rounds up.

Therefore the 3rd digit in the hundredths column is increased from 8 to 9 and 3.3685 rounds up to 3.369.

Here are some examples of rounding to 3 decimal places.

Why do we Round Decimals?

The purpose of rounding decimals is to make a number easier to work with or to read. The further a digit is after the decimal point, the less value it contributes to the overall size of the number. Therefore, removing later digits does not significantly change the value of the number. For example, instead of writing 3.14159, it can be easier to say that this number is approximately 3 or 3.1.

Rounding shortens the length of a decimal.

For example, 3.3685 contains 4 digits after the decimal point. Rounding this number to 3 decimal places we get 3.369, which only has 3 digits after the decimal point.

Many decimal numbers can go on for a very long time, containing lots of decimal places. Some decimal numbers are called irrational numbers and they go on forever.

For example, pi is an irrational number. The first few digits of pi are 3.14159. However, it would be impossible to read the entire number because it goes on forever. For example, adding one more decimal place we get 3.141592 and then we get 3.1415926, and so on.

At some point, with irrational numbers, we have to cut them off at some stage and rounding tells us the rules for doing this accurately.

How to Enlarge a Shape

Enlarging a shape means to change its size. To enlarge a shape, multiply all lengths of the shape by the scale factor. The scale factor tells us how many times larger the shape will be.

For example, enlarging a shape by scale factor 2 means that all of the sides will become 2 times longer.

This rectangle is 1 square wide and 2 squares long.

Multiplying these lengths by the scale factor of 2, the new image will be 2 squares wide and 4 squares long.

Here is an example of multiplying a triangle by scale factor 2.

To enlarge a triangle with sloping sides, it is easiest to enlarge the base and the height.

The base is 2 squares long and the height is 3 squares long.

Multiply these lengths by the scale factor of 2 to find the enlarged shape.

The base is now 4 squares long and the height is 6 squares long.

The triangle can be drawn by connecting the top of the height to each end of the base.

Here is an example of enlarging a shape by scale factor 3. This means that all of the sides will be three times longer.

The original rectangle has width 1 and length 2.

Multiplying these lengths by 3, the new rectangle image has width 3 and length 6.

Here is another example of enlarging a shape by scale factor 3.

The original triangle object has a base of 2 and a height of 3.

Multiplying these lengths by 3, the new triangle image has a base of 6 and a height of 9.

How to Enlarge a Shape by a Scale Factor

To enlarge a shape by a scale factor, first count the distances of each point on the shape from the centre of enlargement. Multiply each distance by the scale factor to find its new position. For example, if a point is 2 to the right and 1 up from the centre of enlargement and the scale factor is 2, the new position will be 4 to the right and 2 up from the centre of enlargement.

For example, enlarge the following triangle by scale factor 3 about the centre of enlargement.

The topmost corner is 3 right and 1 down from the centre of enlargement. Multiply these distances by the scale factor to find the new position of this corner.

3 × 3 = 9 and so, the new corner will be 9 to the right of the centre of enlargement.

1 × 3 = 3 and so, the new corner will be 3 down from the centre of enlargement.

The corner moved from 3 right, 1 down to be 9 right, 3 down.

By multiplying all distances by 3, we see the following distances to the enlarged shape.

Here is another example of enlarging a shape about a point.

This time the scale factor is 2 and so, all distances have been multiplied by 2.

The following table shows the positions of the new corners of the enlarged shape.

Once all of the corners of the enlarged shape have been put in position, the shape can be drawn by connecting the corners.

How to Find the Centre of Enlargement

The centre of enlargement is the coordinate about which a shape is enlarged. To find the centre of enlargement, draw straight lines from each corner of the enlarged shape back through the corners of the original shape. The centre of enlargement is the point where all of the lines cross over.

We will find the centre of enlargement of the example below.

The first step is to find two corresponding corners. We have marked each of the corresponding corners in the same colour.

Use a ruler to draw a straight line between the corner of the enlarged shape and the same corner on the original object.

Continue this straight line far past the original object.

Once you have repeated this for two corners, the two straight lines drawn should cross over. This is the position of the centre of enlargement.

We can check our answer by drawing lines through more of the corners and checking that they all cross over in the same position.

In this example, the centre of enlargement is at (2, 1).

How to Find the Scale Factor of Enlargement

To find the scale factor of enlargement, find two corresponding sides that match up on the original shape and the enlarged shape. Divide the longer side by the shorter side to find the scale factor of enlargement.

For example, in the shape below, two corresponding sides are marked. These are both the rightmost sides of the shape.

One side is 2 squares long and the other side is 6 squares long.

To find the scale factor of enlargement, divide the longer side by the short side.

6 ÷ 2 = 3 and so, the scale factor is 3.

This means that all of the sides on the smaller shape have been multiplied by 3.

It also means that all of the sides on the larger shape can be divided by 3 to make the smaller shape.

We can check this answer by repeating this process for any of the other corresponding sides.

For example, the bottom side of the larger shape is 3 long and the bottom side of the smaller shape is 1 long.

3 ÷ 1 = 3 and this confirms our answer of the scale factor being equal to 3.

What is a One-Step Equation?

A one-step equation is an equation that only involves one mathematical operation to solve. A single addition, subtraction, multiplication or division is used to find the value of the variable. Because only one operation is needed, one-step equations are used to introduce the concept of solving equations.

The four types of one-step equation are:

1. Addition
2. Subtraction
3. Multiplication
4. Division

These solving the 4 different types of one step equations are shown in the image below.



We can see that the variable 𝑥 plus a number can be solved by subtracting this number.

We can see that the variable x subtract a number can be solved by adding this number.

When the variable has a number in front of it, it is being multiplied by that number. We must divide by this number to solve the equation.

When the variable is written in a fraction with a number below it as the denominator, we must multiply by this denominator to solve the equation.

For example, 𝑥 – 8 = 3 is an exampe of a one-step equation. It can be solved in one-step by adding 8 to both sides. 3 + 8 = 11 and so, 𝑥 = 11.



How to Solve One-Step Equations

To solve one-step equations, use the inverse (opposite) of the operation that is acting on the variable. For example, if there is an addition, use subtraction, if there is a subtraction use addition, if there is multiplication, use division and if there is division, use multiplication. This is done to both sides of the equals sign in order to leave the variable on its own on one side of the equals sign.

For example, 𝑥 + 2 = 6 is a one-step equation involving an addition. We will solve it with a subtraction.



The opposite of adding 2 is to subtract 2 to both sides of the equation.

𝑥 + 2 – 2 = 𝑥.

Subtracting 2 has the effect of removing the + 2.

Because we subtracted 2 to the left hand side of the equation, we subtract 2 from the right hand side of the equation. 6 – 2 = 4.

Therefore 𝑥 = 4.

Here is another example of solving a one-step equation.

We have 2𝑥 = 10. Here, 2𝑥 means 2 × 𝑥. The variable 𝑥 is multiplied by 2.

We want to remove the multiplication by 2 to leave just 𝑥. We will do the opposite of multiplying by 2, which is to divide by 2.



2𝑥 ÷ 2 = 𝑥 and 10 ÷ 2 = 5.

Therefore 𝑥 = 5.

How to Solve One-Step Equations with Addition

To solve a one-step equation that contains an addition of a number, subtract this number from both sides of the equation. For example, in 𝑥 + 7 = 15, we remove the + 7 by subtracting 7 from both sides to leave 𝑥 = 8.



On the left hand side, 𝑥 + 7 – 7 = 𝑥.

On the right hand side, 15 – 7 = 8.

We have subtracted 7 from both sides to get 𝑥 = 8.

How to Solve One-Step Equations with Subtraction

To solve a one-step equation that contains the subtraction of a number, add this number from both sides of the equation. For example, in 𝑥 – 2 = 3, add 2 to both sides to get 𝑥 = 5.



On the left hand side, 𝑥 – 2 + 2 = 𝑥. Adding 2 removes the -2.

On the right hand side, 3 + 2 = 5.

The one-step equation is solved to leave 𝑥 = 5.

How to Solve One-Step Equations with Multiplication

A one-step equation involving multiplication will have a number written in front of the variable. To solve this equation, divide both sides of the equation by this number. For example, in 2𝑥 = 10, the variable is multiplied by 2. To remove the 2, divide both sides of the equation by 2 to leave 𝑥 = 5.



2𝑥 means 2 multiplied by 𝑥. To remove the multiplication by 2, divide by 2 to leave 𝑥.

We must also divide the right hand side of the equation by 2. 10 ÷ 2 = 5.

Therefore 𝑥 = 5.

How to Solve One-Step Equations with Fractions

A one-step equation with a fraction involves dividing the variable on top of the fraction by the number on the bottom of the fraction. To solve this equation, multiply both sides of the equation by the number on the bottom of the fraction. For example,   ^𝑥/₉ = 5 can be solved by multiplying both side of the equation by 9 to get 𝑥 = 45.



In the equation   ^𝑥/₉ = 5, the denominator on the bottom of the fraction is 9. This means that 𝑥 is being divided by 9.

To remove the fraction and leave just 𝑥 on its own, we must multiply by 9. This has the effect of cancelling out the division by 9.

Because we multiplied the left-hand side by 9, we also multiply the right-hand side by 9. 5 multiplied by 9 is 45 and so, 𝑥 = 45.

Here is another example of an equation containing a fraction. We have   ^𝑥/₃ = 4.



The denominator of the fraction is 3 and so,   ^𝑥/₃   means 𝑥 ÷ 3.

The inverse of dividing by 3 is to multiply by 3.

On the left hand side of the equation,   ^𝑥/₃   × 3 = 𝑥. This is because dividing by 3 and multiplying by 3 cancel out.

What we do to one side of the equation, we must do to the other and so, we multiply 4 by 3.

4 × 3 = 12 and therefore, 𝑥 = 12.

One-Step Equations with Decimals

Here are some examples of solving one-step equations with decimals.

Here is 𝑥 + 2.4 = 5.5.

To remove the decimal of 2.4, we subtract 2.4 from both sides.



5.5 – 2.4 = 3.1 and so, 𝑥 = 3.1

Here is 𝑥 – 3.6 = 1.2. We have decimals on both sides of this one-step equation.



To remove the subtraction of the decimal – 3.6, we must add 3.6 to both sides.

𝑥 = 4.8.

In the example of 1.5𝑥 = 6, the variable is multiplied by a decimal. We must divide by this decimal to remove it.



Dividing 1.5𝑥 by 1.5 just leaves 𝑥.

To divide 6 by 1.5, we can think how many times 1.5 goes into 6. 1.5 goes into 6 four times and so 𝑥 = 4.

Here we have   ^𝑥/_3.5 = 2. We have a decimal on the denominator of this one-step equation.



We simply need to multiply both sides of the equation by 3.5.

2 × 3.5 = 7 and so, 𝑥 = 7.

What are the Units of Volume?

The most commonly used units of volume from smallest to largest are:

Millitres are used to measure small quantities of liquid, typically less than 1 litre. Examples of items measured in millilitres are medicine containers, drink cartons and shampoo.

Centilitres are used to measure small quantities of liquid such as wine in a glass and engine capacity. Centilitres are used less frequently to measure capacity than millilitres or litres are.

Litres are used to measure most quantities of liquid that are greater than a large drink bottle. Examples of items measured in litres are bottles of milk, petrol in a car and buckets of water.

Kilolitres are used to measure very large quantities of liquid such as oil in a storage tank, water in a fire truck and the amount of water used for the water bill of a household.

How Much is 1 Millilitre?

1 millilitre is the size of a droplet of rain. 5 millilitres fit on a level teaspoon and 250 millilitres is the same amount in a small carton of juice. 1000 millilitres make one litre.

Millilitres is written as milliliters in the USA. Millilitres is written as mL or ml for short.

1 mL is the amount of water in a single droplet.

Millilitres are often used to measure liquid in measuring flasks. Below are three different flasks showing 30 mL, 50 mL and 100 mL.

30 mL is about the size of one mouthful. 100 mL is the size of a small tube of toothpaste.

A carton of juice contains 250 mL. A can of soft drink contains 375 mL.

A typical bottle of drink that is for one person to consume contains 500 mL.

How Much is 1 Litre?

1 litre is equivalent to 1000 millilitres and it is the amount of liquid in a large water bottle. 2 litres is enough to fill a kettle to the brim and there are about 100 litres in a bathtub.

1000 millilitres make 1 litre.

1 litre is approximately the size of a large drink bottle that can be held in one hand. It is twice as much as the typical drink bottle sizes sold in vending machines.

The largest bottles of drinks that can be bought in supermarkets contain 2 litres. 2L is the amount found in the family size drink bottles.

Litres are used to measure most quantities of capacity larger than 1 litre.

For example, filling a car requires about 50 litres.

A full bathtub contains approximately 150 litres of water.

A wine barrel is an example of something measured in litres. There are approximately 200 litres of wine in a wine barrel.

How Much is 1 Kilolitre?

1 kilolitre is as large as 1000 litres. 1 kilolitre is as much water as 8 full bathtubs. There are 2,500 kilolitres of water in an olympic sized swimming pool.

Typical water bills measure the amount of water used by a household each time period in kilolitres. The price of water is given in cost per kilolitre.

How to Convert Units of Volume

To convert between metric units of volume, use the following rules:

Kilo means 1000. Therefore 1 kilolitre contains 1000 litres. To convert from kilolitres to litres, simply multiply the amount by 1000. For example, 3 kL = 3000 L.

To convert from L to kL, divide the value by 1000. For example, 257 L = 0.257 kL.

Milli means one-thousandth. Therefore 1 litre contains 1000 millilitres. To convert from litres to millilitres, simply multiply the amount by 1000. For example, 5 L = 5000 mL.

To convert from mL to L, divide the value by 1000. For example, 375 mL = 0.375 L.

Choosing Appropriate Units of Volume

The most common units of volume are millilitres and litres. Millilitres are chosen to measure volumes less than 1 litre (about the size of a large bottle). Litres are chosen to measure volumes equal to 1 litre or greater.

As a general rule, if the amount being measured is less than the size of a large bottle of water, it should be measured in mL. If the amount is larger than a large bottle, it should be measured in L.

For example, a glass of water, a can of drink and a dose of medicine are all measured in millilitres because they are all less than 1 litre.

It is better to say that a can is 375 mL instead of 0.375 L. This is because it is easier to visualise how much a whole number is compared to a decimal number. Each can contains 375 millilitre drops.

Filling up a car is measured in litres because it is a relatively large amount of volume. It is better to find whole number values that are not too large (larger than 1000).

It is better to say that the car contains 50 L instead of 50, 000 mL. We are also filling the car quickly, so we can measure each litre as it fills the car. It would be too difficult to fill the car millilitre by millilitre as it comes out too quickly.

Similarly, a barrel of oil would be measured in litres as it is larger than a large bottle. It is better to say that the barrel contains 200 L instead of 200, 000 mL.

What are Unlike Fractions?

Unlike fractions are two or more fractions that have different denominators. This means that the numbers on the bottom of unlike fractions are different. Unlike fractions are each divided into different-sized parts and so, they cannot be compared or added easily.

For example,   ¹ / ₄   and   ¹ / ₃   are unlike fractions because the denominators are different. The first fraction is divided into 3 parts and the second fraction is divided into 4 parts.

Like fractions are fractions that have the same denominator on the bottom. For example,   ¹ / ₄   and   ³ / ₄  . These fractions both have the same denominator of 4.

We can see that   ³ / ₄   is larger than   ¹ / ₄   because we have 3 quarters compared to 1 quarter. We are comparing parts that are the same size.

However, it is not immediately clear which is larger out of the unlike fractions of   ² / ₃   and   ³ / ₅  .

How to Compare Fractions with Unlike Denominators

To compare fractions with different denominators, follow these steps:

1. Find the first number to appear in the times table of every denominator. This is the common denominator.
2. Write each fraction as an equivalent fraction that has this common denominator.
3. The larger the numerator of these fractions, the larger the fraction.

As an example, we will compare the unlike fractions of   ¹ / ₄   and   ² / ₆  .

Step 1 is to find the first number to appear in the times table of both denominators. The 4 times table is 4, 8, 12, 16, 20, 24. The 6 times table is 6, 12, 18, 24.

We only need to list the numbers up to 24 because 4 × 6 = 24. However We can see that 12 is a number that appears first. We will use 12 as the common denominator.

Step 2 is to write both fractions as equivalent fractions with a common denominator. We will write both fractions out of 12.

  ¹ / ₄   =   ³ / ₁₂   and   ² / ₆   =   ⁴ / ₁₂  .

Step 3 is to compare the fractions using their numerator.

3 is less than 4 and so,   ⁴ / ₁₂   is less than   ³ / ₁₂  .

This allows us to compare the original fractions.

  ¹ / ₄   is less than   ² / ₆  .

We can write this as   ¹ / ₄   <   ² / ₆  .

We can use the inequality signs of greater less and less than to compare fractions. The sign of ‘<' or '>‘ always opens up to face the larger fraction and points at the smaller fraction.

Here is another example of comparing fractions with different denominators. We have   ³ / ₄   and   ⁷ / ₈  .

We can see that 8 is the first number in the 4 and 8 times table. We double the numbers in the first fraction so that   ³ / ₄   =   ⁶ / ₈  .

We can keep   ⁷ / ₈   the same.

Now that both fractions are out of 8, we can compare them.

6 is less than 7 and so,   ⁶ / ₈   is less than   ⁷ / ₈  .

Therefore we can say that   ³ / ₄   is less than   ⁷ / ₈  .

We write   ³ / ₄   <   ⁷ / ₈  . The arrow of the sign points to the smaller fraction.

How to Compare Improper Fractions with Different Denominators

To compare improper fractions with different denominators, first write each fraction as its equivalent fraction so that they both have the same denominator. The largest improper fraction will now have the largest numerator. Two improper fractions can only be compared if they have the same denominator on the bottom.

For example, here are the improper fractions of   ⁵ / ₄   and   ⁶ / ₅  . Improper fractions are simply fractions that have a larger numerator on top than their denominator on the bottom.

The common denominator of these fractions is 20. 20 is the first number in both the 4 and 5 times table.

Writing the fractions as equivalent fractions with denominators of 20,   ⁵ / ₄   =   ²⁵ / ₂₀   and   ⁶ / ₅   =   ²⁴ / ₂₀  .

25 is a larger numerator than 24 and so,   ²⁵ / ₂₀   is larger than   ²⁴ / ₂₀  .

Therefore   ⁵ / ₄   is greater than   ⁶ / ₅  .

Here is another example of comparing the size of two improper fractions.

We have   ¹⁰ / ₃   and   ⁹ / ₂  .

We write   ¹⁰ / ₃   =   ²⁰ / ₆   and   ⁹ / ₂   =   ²⁷ / ₆  .

Both fractions have the same denominator and can now be ordered.

20 is less than 27 and so,   ²⁰ / ₆   is less than   ²⁷ / ₆  .

Therefore   ¹⁰ / ₃   is less than   ⁹ / ₂  .

Comparing Unlike Fractions using Decimals

Converting unlike fractions to decimals is a method that can be used to compare the size of them. Fractions can be turned into decimals by dividing the numerator by the denominator. The larger the decimal number, the larger the fraction.

For example, here we have the fractions   ² / ₅   and   ³ / ₄  .

We turn   ² / ₅   into a decimal by dividing 2 by 5.

2 ÷ 5 = 0.4.

We turn   ³ / ₄   into a decimal by dividing 3 by 4.

3 ÷ 4 = 0.75.

The larger decimal number is 0.75. Therefore the larger fraction is   ³ / ₄  .

We say that   ² / ₅   is less than   ³ / ₄