Ratio

Key Concepts

• A ratio compares two or more quantities.

  • It shows how much of one quantity there is compared with another.
  • Ratios are written using a colon, for example 3 : 5.
  • This means “3 parts to 5 parts”.

• A ratio can compare:

  • part to part
    Example: 2 red balls and 3 blue balls → ratio of red to blue = 2 : 3
  • part to whole
    Example: 2 red balls and 3 blue balls, total = 5 balls → ratio of red balls to total balls = 2 : 5

• The order in a ratio is very important.

  • Ratio of boys to girls is not the same as ratio of girls to boys.
  • Example:
    • boys : girls = 2 : 3
    • girls : boys = 3 : 2

• Ratios can be written in equivalent forms.

  • Equivalent ratios name the same comparison.
  • Example:
    • 1 : 2
    • 2 : 4
    • 3 : 6
      These are all equivalent because both sides are multiplied by the same number.

• A ratio should usually be given in its simplest form.

  • This means the numbers in the ratio have no common factor other than 1.
  • Example:
    • 8 : 12
    • divide both terms by 4
    • simplest form = 2 : 3

• To simplify a ratio:

  1. Write the ratio clearly.
  2. Find the greatest common factor (GCF) or a common factor of both numbers.
  3. Divide both terms by the same number.

• A ratio is different from a fraction, but they are related.

  • Ratio compares quantities.
  • Fraction shows a part of a whole.
  • Example:
    • ratio of red to blue = 2 : 3
    • fraction of red out of total = 2 out of 5 = 2/5

• When dividing a quantity in a given ratio:

  • First find the total number of parts.
  • Then divide the total quantity by the total number of parts.
  • This gives the value of 1 part.
  • Then multiply to find each share.

• In before and after ratio problems:

  • Something may be added, removed, eaten, spent, or transferred.
  • You must be careful about:
    • what changed
    • what stayed the same
    • whether the total changed
  • It often helps to use units or a bar model.

• In some ratio questions:

  • The actual amount is unknown.
  • Use equal units to represent the quantities.
  • Then compare the “before” and “after” situations carefully.

• Ratios can only compare quantities with the same unit.

  • Example:
    • 200 g : 500 g = 2 : 5
  • If the units are different, convert first.
    • Example:
      • 1 m : 50 cm
      • convert 1 m to 100 cm
      • 100 : 50 = 2 : 1

Three-Term Ratios

• A three-term ratio compares three quantities using two colons.

  • Example: A : B : C = 2 : 3 : 5 means A gets 2 parts, B gets 3 parts and C gets 5 parts.

• To simplify a three-term ratio:

  • Find the HCF of all three numbers.
  • Divide all three terms by the HCF.
  • Example:
    • 4 : 6 : 10
    • HCF of 4, 6 and 10 = 2
    • Divide each term by 2: 2 : 3 : 5

• To divide a quantity in a three-term ratio:

  1. Add all the parts to find the total number of parts.
  2. Divide the total quantity by the total parts to find 1 part.
  3. Multiply to find each share.
  • Example: “Share $100 in the ratio 2 : 3 : 5.”
    • Total parts = 2 + 3 + 5 = 10
    • 1 part = $100 ÷ 10 = $10
    • A gets 2 × $10 = $20
    • B gets 3 × $10 = $30
    • C gets 5 × $10 = $50

• To combine two ratios into a three-term ratio:

  • Make the shared quantity (B) the same number in both ratios.
  • Then read off A : B : C.
  • Example:
    • A : B = 1 : 2 and B : C = 3 : 4
    • Multiply A : B by 3 → A : B = 3 : 6
    • Multiply B : C by 2 → B : C = 6 : 8
    • Now B is 6 in both, so A : B : C = 3 : 6 : 8

Worked Example: Dividing in a Three-Term Ratio

$180 is shared among Ali, Ben and Carol in the ratio 1 : 2 : 3. How much does each person get?

Step 1: Find the total number of parts

• 1 + 2 + 3 = 6 parts

Step 2: Find the value of 1 part

• $180 ÷ 6 = $30

Step 3: Find each person’s share

• Ali gets 1 part: 1 × $30 = $30
• Ben gets 2 parts: 2 × $30 = $60
• Carol gets 3 parts: 3 × $30 = $90

Final Answer

• Ali gets $30, Ben gets $60, Carol gets $90.

Check

• $30 + $60 + $90 = $180 ✔
• Ratio: 30 : 60 : 90 = 1 : 2 : 3 ✔

Important Definitions

• Ratio: a comparison of two or more quantities using division.

• Term of a ratio: each number in a ratio.
  Example: in 3 : 4, the terms are 3 and 4.

• Equivalent ratios: ratios that show the same comparison even though the numbers are different.
  Example: 1 : 2 and 3 : 6.

• Simplest form: a ratio written using the smallest whole numbers possible.

• Common factor: a number that divides exactly into two or more numbers.

• Greatest common factor (GCF): the largest number that divides exactly into both terms of a ratio.

• Part-part comparison: a ratio comparing one part of a group to another part of the same group.
  Example: apples to oranges.

• Part-whole comparison: a ratio comparing one part of a group to the total number in the group.
  Example: apples to all fruits.

• Total parts: the sum of the terms in a ratio.
  Example: in 2 : 5, total parts = 2 + 5 = 7.

• Unit value: the value of 1 part when a total quantity is divided according to a ratio.

Worked Examples

Example 1: Equivalent Ratios and Simplest Form

The ratio of pencils to pens is 12 : 18. Write the ratio in simplest form.

Step 1: Find a common factor

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Greatest common factor = 6

Step 2: Divide both terms by 6

• 12 ÷ 6 = 2
• 18 ÷ 6 = 3

Step 3: Write the simplified ratio

• Ratio in simplest form = 2 : 3

Check

• 2 : 3, 4 : 6, 6 : 9, 12 : 18 are all equivalent ratios.

Example 2: Dividing a Quantity in a Given Ratio

A sum of $72 is divided between Sarah and Mei in the ratio 5 : 4. How much does each girl get?

Step 1: Find total parts

• 5 + 4 = 9 parts

Step 2: Find the value of 1 part

• $72 ÷ 9 = $8

Step 3: Find each share

• Sarah gets 5 parts:
  • 5 × $8 = $40
• Mei gets 4 parts:
  • 4 × $8 = $32

Final Answer

• Sarah gets $40
• Mei gets $32

Check

• $40 + $32 = $72 ✔

Example 3: Before and After Ratio Problem

The ratio of red beads to blue beads was 3 : 5. After 12 blue beads were removed, the ratio became 3 : 2. How many blue beads were there at first?

Step 1: Understand what stayed the same

• The number of red beads stayed the same.
• Only blue beads changed.

Step 2: Compare the two ratios

Before:

• red : blue = 3 : 5

After:

• red : blue = 3 : 2

Since the red part is 3 in both ratios, we can compare directly.

Step 3: Find the change in blue parts

• Blue beads changed from 5 parts to 2 parts.
• Difference = 5 − 2 = 3 parts

We are told this difference is 12 blue beads.

Step 4: Find the value of 1 part

• 12 ÷ 3 = 4 beads

Step 5: Find the original number of blue beads

• At first, blue beads = 5 parts
• 5 × 4 = 20 beads

Final Answer

• There were 20 blue beads at first.

Check

• Red beads = 3 parts = 3 × 4 = 12
• Before: 12 : 20 = 3 : 5 ✔
• After removing 12 blue beads:
  • blue = 20 − 12 = 8
• New ratio:
  • 12 : 8 = 3 : 2 ✔

Common Mistakes to Avoid

• Writing the ratio in the wrong order.

  • If the question asks for cats to dogs, do not write dogs to cats.

• Forgetting to simplify the ratio.

  • Example: writing 4 : 8 instead of 1 : 2.

• Dividing only one term when simplifying.

  • You must divide both terms by the same number.

• Adding ratio terms wrongly.

  • For 2 : 3, total parts is 5, not 6.

• Mixing up part-part and part-whole ratios.

  • If red : blue = 2 : 3, then red : total is 2 : 5, not 2 : 3.

• Using different units without converting first.

  • Example: do not compare 1 kg and 500 g directly.
  • Convert 1 kg to 1000 g first.

• In before-and-after questions, changing the wrong quantity.

  • Read carefully to know what was added, removed, or unchanged.

• Assuming “3 parts” always means 3 items.

  • A part can stand for many items.

• Forgetting to check whether the total quantity changed.

  • If something is added or removed, the total may not stay the same.

• Not checking the final answer against the new ratio.

Exam Tips

• Underline key words in the question:

  • to
  • out of
  • total
  • remaining
  • shared
  • simplest form
  • before
  • after

• For ratio sharing questions, always write:

  1. Total parts =
  2. 1 part =
  3. Required amount =

• Use a bar model for word problems.

  • It helps you avoid guessing.

• If the question says “in the ratio”, the numbers are parts, not actual amounts unless stated.

• If the question asks for part to whole, remember:

  • whole = sum of all parts

• In before-and-after questions:

  • Identify what did not change.
  • That unchanged quantity helps you match the ratios.

• Always give ratios in simplest form unless the question says otherwise.

• Do a quick check:

  • Does the answer match the total?
  • Does the answer fit the ratio?
  • Is the order correct?

• Good mark-earning working should include clear statements such as:

  • Total parts = 3 + 4 = 7
  • 1 part = 56 ÷ 7 = 8
  • Therefore, …
  • Check: …

Quick Summary

• A ratio compares two or more quantities.
• The order of the terms in a ratio matters.
• Equivalent ratios are made by multiplying or dividing both terms by the same number.
• Write ratios in simplest form using the greatest common factor.
• Part-part ratio compares one part with another part.
• Part-whole ratio compares one part with the total.
• For part-whole questions, total = sum of all parts.
• To divide a quantity in a ratio:
  • find total parts
  • find 1 part
  • multiply to get each share
• In before-and-after problems, look for the quantity that stays the same.
• Convert to the same units before forming or simplifying a ratio.
• Use a bar model to organise information clearly.
• Always check that your final answer matches the ratio and the total.