UK K12 GCSE/A-Level: Year 7 KS3 Mathematics - Ratio and Proportion

Learning Objectives

By the end of this topic, students will be able to: - Define ratio and proportion in mathematical terms - Identify and write ratios in simplest form - Compare ratios using equivalence and inequality - Apply ratio and proportion to solve problems in real-world contexts - Use ratio and proportion to scale quantities and shapes - Identify and explain common misconceptions related to ratio and proportion

Core Concepts

What is a Ratio?

A ratio is a comparison of two or more numbers. It is often written as a fraction, with the first number as the numerator and the second number as the denominator. For example, the ratio of boys to girls in a class of 20 students can be written as 10:10 or 1:1.

What is a Proportion?

A proportion is a statement that two ratios are equal. For example, if the ratio of boys to girls in a class is 1:1, then the proportion 10:10 = 20:20 is true.

Equivalent Ratios

Equivalent ratios are ratios that have the same value, but are written in different forms. For example, the ratios 2:4, 3:6, and 4:8 are all equivalent because they can be simplified to 1:2.

Scaling Quantities and Shapes

When we scale a quantity or shape, we are changing its size while keeping its proportions the same. For example, if we have a rectangle with a length of 5 cm and a width of 3 cm, we can scale it up by a factor of 2 to get a new rectangle with a length of 10 cm and a width of 6 cm.

Worked Examples

Example 1: Writing Ratios in Simplest Form

A recipe for making cookies calls for a ratio of 2 cups of flour to 1 cup of sugar. Write this ratio in simplest form.

To simplify the ratio, we need to find the greatest common divisor (GCD) of 2 and 1, which is 1. We can then divide both numbers by the GCD to get a new ratio of 2:1.

Example 2: Comparing Ratios

A bakery has two types of bread: whole wheat and white. The ratio of whole wheat bread to white bread is 3:2. If the bakery sells 18 loaves of whole wheat bread, how many loaves of white bread does it sell?

To solve this problem, we need to find the number of white bread loaves that corresponds to 18 whole wheat loaves. We can do this by setting up a proportion:

3/2 = 18/x

We can then cross-multiply and solve for x to get x = 12.

Example 3: Scaling Quantities

A bookshelf has a length of 5 meters and a height of 2 meters. If we want to scale it up by a factor of 3, what will be the new dimensions of the bookshelf?

To solve this problem, we can multiply both the length and height of the bookshelf by 3 to get new dimensions of 15 meters and 6 meters.

Common Misconceptions

• Many students think that equivalent ratios must have the same numbers, but this is not true. Equivalent ratios can have different numbers as long as they have the same value.
• Some students think that scaling a quantity or shape means changing its proportions, but this is not true. Scaling a quantity or shape means changing its size while keeping its proportions the same.
• Many students think that proportions must be equal, but this is not true. Proportions can be equal or unequal, depending on the context.

Exam Tips

• Make sure to read the question carefully and understand what is being asked.
• Use equivalent ratios to simplify complex ratios.
• Use proportions to solve problems involving ratios.
• Be careful when scaling quantities and shapes to make sure you keep the proportions the same.
• Check your answers to make sure they make sense in the context of the problem.

MCQs with Explanations

MCQ 1: [F]

What is the ratio of 6:8 in simplest form?

A) 1:2 B) 2:3 C) 3:4 D) 6:8

Correct answer: A) 1:2

Why the distractors fail: Options B, C, and D are all incorrect because they do not simplify the ratio to its simplest form.

MCQ 2: [H]

If the ratio of boys to girls in a class is 3:5, what is the proportion 15:25 = ?

A) 3:5 B) 5:3 C) 15:25 D) 25:15

Correct answer: A) 3:5

Why the distractors fail: Options B, C, and D are all incorrect because they do not represent the correct proportion.

MCQ 3: [F]

A recipe for making cookies calls for a ratio of 2 cups of flour to 1 cup of sugar. If we want to make half the recipe, what is the new ratio of flour to sugar?

A) 1:1 B) 2:1 C) 1:2 D) 1:0.5

Correct answer: C) 1:2

Why the distractors fail: Options A, B, and D are all incorrect because they do not represent the correct ratio.

MCQ 4: [H]

A bookshelf has a length of 5 meters and a height of 2 meters. If we want to scale it up by a factor of 2, what will be the new dimensions of the bookshelf?

A) 10 meters and 4 meters B) 5 meters and 2 meters C) 20 meters and 4 meters D) 10 meters and 1 meter

Correct answer: A) 10 meters and 4 meters

Why the distractors fail: Options B, C, and D are all incorrect because they do not represent the correct dimensions.

MCQ 5: [H]

A bakery has two types of bread: whole wheat and white. The ratio of whole wheat bread to white bread is 3:2. If the bakery sells 18 loaves of whole wheat bread, how many loaves of white bread does it sell?

A) 12 loaves B) 15 loaves C) 18 loaves D) 24 loaves

Correct answer: A) 12 loaves

Why the distractors fail: Options B, C, and D are all incorrect because they do not represent the correct number of white bread loaves.

Short-answer Questions

Question 1

A recipe for making cookies calls for a ratio of 2 cups of flour to 1 cup of sugar. Write this ratio in simplest form.

Question 2

A bakery has two types of bread: whole wheat and white. The ratio of whole wheat bread to white bread is 3:2. If the bakery sells 18 loaves of whole wheat bread, how many loaves of white bread does it sell?

Question 3

A bookshelf has a length of 5 meters and a height of 2 meters. If we want to scale it up by a factor of 3, what will be the new dimensions of the bookshelf?

Question 4

A recipe for making cookies calls for a ratio of 2 cups of flour to 1 cup of sugar. If we want to make half the recipe, what is the new ratio of flour to sugar?

Question 5

A bakery has two types of bread: whole wheat and white. The ratio of whole wheat bread to white bread is 3:2. If the bakery sells 12 loaves of white bread, how many loaves of whole wheat bread does it sell?