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GCSE Edexcel Maths › Ratios (GCSE Maths)

Ratios (GCSE Maths)

Skill Check

Q1
Question 1

Alex and Bob share money in the ratio 3 : 8.

Bob receives £35 more than Alex.

How much money do they share in total?

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SOLUTION

Step 1  ·  Find the difference in ratio parts

The ratio of Alex's share to Bob's share is $3 : 8$.

$$8 - 3 = 5\text{ parts}$$

Step 2  ·  Find the value of a single part

  • We are told that this difference of $5$ parts is equal to $£35$.
  • To find the value of $1$ part, divide the monetary difference by the difference in parts.
$$£35 \div 5 = £7\text{ per part}$$

Step 3  ·  Find the total number of parts

Add the numbers in the ratio together to find the total shares.

$$3 + 8 = 11\text{ total parts}$$

Step 4  ·  Calculate the total amount of money

Multiply the total number of parts by the value of a single part.

$$11 \times £7 = £77$$

Final Answer

$£77$

Q2
Question 2

A piece of wood is 240 cm long.

It is cut into three pieces in the ratio 2 : 3 : 7.

How long is the shortest piece?

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to find out how many equal parts the wood is divided into in total.

$$2 + 3 + 7 = 12$$

Step 2  ·  Find the length of a single part

Divide the total length of the wood by the total number of parts.

$$240 \div 12 = 20\text{ cm}$$

Step 3  ·  Calculate the length of the shortest piece

  • The shortest piece corresponds to the smallest number in the ratio, which is $2$.
  • Multiply the length of one part by $2$.
$$20 \times 2 = 40\text{ cm}$$

Final Answer

$40\text{ cm}$

Q3
Question 3

The angles in a triangle are in the ratio 1 : 2 : 3.

Calculate the size of the largest angle.

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SOLUTION

Step 1  ·  State the total sum of angles

The interior angles of any triangle always add up to $180^\circ$.

Step 2  ·  Find the total number of parts in the ratio

Add the numbers in the ratio together to find the total number of shares.

$$1 + 2 + 3 = 6$$

Step 3  ·  Find the value of a single part

Divide the total degrees by the total number of parts.

$$180^\circ \div 6 = 30^\circ$$

Step 4  ·  Calculate the largest angle

  • The largest angle corresponds to the largest number in the ratio, which is $3$.
  • Multiply the value of one part by $3$.
$$30^\circ \times 3 = 90^\circ$$

Final Answer

$90^\circ$

Q4
Question 4

The ratio of cats to dogs in a shelter is 4 : 5.

There are 24 cats.

How many dogs are there?

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SOLUTION

Step 1  ·  Identify the parts of the ratio

  • The ratio of cats to dogs is given as $4 : 5$.
  • This means for every $4$ parts of cats, there are $5$ parts of dogs.

Step 2  ·  Find the value of a single part

We are told there are $24$ cats, which corresponds to the $4$ parts in our ratio. To find the value of $1$ part, divide the total number of cats by $4$.

$$24 \div 4 = 6$$

So, $1\text{ part} = 6\text{ animals}$.

Step 3  ·  Calculate the number of dogs

The dogs represent $5$ parts of the ratio. Multiply the value of a single part by $5$.

$$6 \times 5 = 30$$

Final Answer

$30\text{ dogs}$

Q5
Question 5

Simplify the ratio 45 : 60.

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SOLUTION

Step 1  ·  Find a common factor

  • Look for the highest number that divides evenly into both $45$ and $60$.
  • The Highest Common Factor (HCF) is $15$.
  • (Tip: If you don't spot $15$ right away, you can start by dividing both numbers by a smaller common factor like $5$ or $3$, and then repeat the process with the new numbers).

Step 2  ·  Divide both sides by the common factor

$$45 \div 15 = 3$$
$$60 \div 15 = 4$$

Final Answer

$3 : 4$

Problem Solving

Q6
Question 6

Divide 1440 in the ratio 3 : 5 : 4.

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the total amount is divided into:

$$3 + 5 + 4 = 12$$

Step 2  ·  Find the value of a single part

Divide the total amount ($1440$) by the total number of parts ($12$):

$$1440 \div 12 = 120$$

Step 3  ·  Calculate the value for each part

Multiply the value of one part by each number in the original ratio:

  • First part: $3 \times 120 = 360$
  • Second part: $5 \times 120 = 600$
  • Third part: $4 \times 120 = 480$

Final Answer

$360,\ 600,\ 480$

Q7
Question 7

Divide 1200 in the ratio 1 : 2 : 3.

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the total amount is divided into:

$$1 + 2 + 3 = 6 \text{ total parts}$$

Step 2  ·  Find the value of a single part

Divide the total amount ($1200$) by the total number of parts ($6$):

$$1200 \div 6 = 200$$

Step 3  ·  Calculate the value for each part

Multiply the value of one part by each number in the original ratio:

  • First part: $1 \times 200 = 200$
  • Second part: $2 \times 200 = 400$
  • Third part: $3 \times 200 = 600$

Final Answer

$200,\ 400,\text{ and } 600$

Q8
Question 8

Divide 840 in the ratio 3 : 4 : 5.

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the total amount is divided into.

$$3 + 4 + 5 = 12$$

Step 2  ·  Find the value of a single part

Divide the total amount ($840$) by the total number of parts ($12$).

$$840 \div 12 = 70$$

Step 3  ·  Calculate the value for each part

Multiply the value of one part by each number in the original ratio:

  • First part: $3 \times 70 = 210$
  • Second part: $4 \times 70 = 280$
  • Third part: $5 \times 70 = 350$

Final Answer

$210, 280, \text{ and } 350$

Q9
Question 9

Divide 450 in the ratio 4 : 5.

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the total amount is divided into.

$$4 + 5 = 9 \text{ parts}$$

Step 2  ·  Find the value of a single part

Divide the total amount ($450$) by the total number of parts ($9$).

$$450 \div 9 = 50$$

Step 3  ·  Calculate the value for each side of the ratio

Multiply the value of one part by each number in the original ratio.

$$\text{First part} = 4 \times 50 = 200$$
$$\text{Second part} = 5 \times 50 = 250$$

Final Answer

$200 \text{ and } 250$

Q10
Question 10

Divide 64 in the ratio 5 : 3.

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SOLUTION

Step 1  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the total amount is divided into.

$$5 + 3 = 8\text{ parts}$$

Step 2  ·  Find the value of a single part

Divide the total number ($64$) by the total number of parts ($8$).

$$\frac{64}{8} = 8$$

Step 3  ·  Calculate the value for each side

Multiply the value of one part by each number in the original ratio.

$$\text{First part} = 5 \times 8 = 40$$
$$\text{Second part} = 3 \times 8 = 24$$

Final Answer

$40\text{ and }24$

Exam-Style Questions

Q11
Question 11

A business makes a profit of £4,500.

It donates 20% of the profit to charity, and the two partners split the remaining profit in the ratio 5 : 4.

How much does the partner with the larger share receive?

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SOLUTION

Step 1  ·  Calculate the remaining profit

First, find out how much is donated to charity. $20\%$ of $£4{,}500$ is $£900$:

$$4{,}500 \times 0.20 = 900$$

Subtract this from the total profit to find the remaining amount to be split:

$$4{,}500 - 900 = 3{,}600$$

(Alternatively, you can just calculate $80\%$ of $£4{,}500$ directly).

Step 2  ·  Find the total number of parts

Add the numbers in the ratio together to determine how many equal parts the remaining profit is divided into:

$$5 + 4 = 9$$

Step 3  ·  Find the value of a single part

Divide the remaining profit ($£3{,}600$) by the total number of parts ($9$):

$$\frac{3{,}600}{9} = 400$$

Step 4  ·  Calculate the larger share

  • The partner with the larger share corresponds to the number $5$ in the ratio.
  • Multiply the value of one part by $5$:
$$400 \times 5 = 2{,}000$$

Final Answer

$£2{,}000$

Q12
Question 12

In a sports club, the ratio of swimmers to runners is 3 : 4, and the ratio of runners to cyclists is 2 : 5.

What is the ratio of swimmers to runners to cyclists?

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SOLUTION

Step 1  ·  Identify the common group

The "runners" appear in both given ratios, linking the swimmers and the cyclists together.

Step 2  ·  Line up the ratios

Look at the numbers representing the runners in both ratios:

$$\text{Swimmers} : \text{Runners} = 3 : 4$$
$$\text{Runners} : \text{Cyclists} = 2 : 5$$

Step 3  ·  Make the common parts equal

Find a common multiple for the runner parts ($4$ and $2$). The lowest common multiple is $4$.

  • The first ratio already has $4$ for runners, so it stays the same ($3 : 4$).
  • To make the runners $4$ in the second ratio, multiply both sides of that ratio by $2$:
$$(2 \times 2) : (5 \times 2) = 4 : 10$$

Step 4  ·  Combine the ratios

Now that the runners are represented by $4$ in both, you can merge them into a single three-part ratio:

$$\text{Swimmers} : \text{Runners} : \text{Cyclists} = 3 : 4 : 10$$

Final Answer

$3 : 4 : 10$

Q13
Question 13

A bowl contains apples and oranges in the ratio 2 : 3.

If 4 more apples are added to the bowl, the new ratio of apples to oranges becomes 1 : 1.

How many oranges are in the bowl?

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SOLUTION

Step 1  ·  Set up the initial ratio with a variable

Let the initial number of apples be $2x$ and the number of oranges be $3x$.

Step 2  ·  Create an equation based on the change

  • We are told that 4 apples are added, so the new amount of apples is $2x + 4$.
  • The number of oranges stays the same at $3x$.
  • The new ratio of apples to oranges is $1 : 1$, which means the number of apples now equals the number of oranges.

We can write this as an equation:

$$2x + 4 = 3x$$

Step 3  ·  Solve for $x$

Subtract $2x$ from both sides of the equation to find the value of $x$:

$$4 = x$$

Step 4  ·  Calculate the number of oranges

  • The question asks for the number of oranges, which is represented by $3x$.
  • Substitute $x = 4$ into the expression:
$$3 \times 4 = 12$$

Final Answer

$12\text{ oranges}$

Q14
Question 14

A landscaper mixes sand, gravel, and cement in the ratio 4 : 5 : 2.

If they use 150 kg of gravel for a project, how much sand and cement do they need?

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SOLUTION

Step 1  ·  Identify the parts of the ratio

  • The ratio of sand to gravel to cement is $4 : 5 : 2$.
  • This means gravel represents $5$ parts of the total mixture.

Step 2  ·  Find the value of a single part

  • We are told that the $5$ parts of gravel equal $150\text{ kg}$.
  • To find the value of $1$ part, divide the total weight of the gravel by its corresponding ratio number.
$$150 \div 5 = 30\text{ kg}$$

Step 3  ·  Calculate the amount of sand

  • Sand represents $4$ parts of the ratio.
  • Multiply the value of a single part by $4$.
$$30 \times 4 = 120\text{ kg}$$

Step 4  ·  Calculate the amount of cement

  • Cement represents $2$ parts of the ratio.
  • Multiply the value of a single part by $2$.
$$30 \times 2 = 60\text{ kg}$$

Final Answer

$120\text{ kg}\text{ sand, } 60\text{ kg}\text{ cement}$

Q15
Question 15

The ratio of adult tickets to child tickets sold at a theme park is 5 : 2.

If 120 more adult tickets were sold than child tickets,

what is the total number of tickets sold?

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SOLUTION

Step 1  ·  Find the difference in ratio parts

The ratio of adult to child tickets is $5 : 2$.

$$5 - 2 = 3\text{ parts}$$

Step 2  ·  Find the value of a single part

  • We are told that this difference of $3$ parts is equal to $120$ tickets.
  • To find the value of $1$ part, divide the numerical difference by the difference in parts.
$$120 \div 3 = 40\text{ tickets per part}$$

Step 3  ·  Find the total number of parts

Add the numbers in the ratio together to find the total shares.

$$5 + 2 = 7\text{ total parts}$$

Step 4  ·  Calculate the total number of tickets

Multiply the total number of parts by the value of a single part.

$$7 \times 40 = 280$$

Final Answer

$280\text{ tickets}$

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