Dividing Quantities in a Given Ratio

In our last lesson, we mastered an efficient method for calculating percentage increases and decreases using decimal multipliers. That was our first deep dive into proportional reasoning. Today, we continue exploring this theme by looking at another common way quantities are related: ratios.

This lesson will teach you how to divide a quantity in a given part-to-part ratio. You'll find this skill is useful in many situations, from mixing ingredients for a recipe to sharing money or items fairly among a group of people. It’s a foundational concept that builds on your understanding of fractions and multiplication.

1. What is a Ratio?

Before we start dividing, let's be clear on what a ratio is. A ratio is a way of comparing quantities. Specifically, it's a part-to-part comparison.

Imagine you're making a drink by mixing 1 part cordial with 4 parts water.

• The ratio of cordial to water is written as 1:4.
• This tells you the relationship between the two ingredients: for every 1 unit of cordial, you need 4 units of water.

This is different from a fraction, which is a part-to-whole comparison. In our drink example, the "whole" mixture has $1+4=5$ parts in total.

• The fraction of cordial in the drink is $\frac{1}{5}$.
• The fraction of water in the drink is $\frac{4}{5}$.

Understanding this difference is the first key step. The following resource from the New Zealand curriculum website explains this distinction clearly using the example of mixing juice.

2. The Method for Sharing in a Ratio

Now that we're clear on what a ratio is, how do we use one to divide a total amount? There is a simple, three-step method that always works.

1. Sum: Add the numbers in the ratio to find the total number of parts.
2. Divide: Divide the total quantity you are sharing by the total number of parts. This tells you the value of a single part.
3. Multiply: Multiply the value of one part by each number in the ratio. This gives you the final amount for each share.

Let's see this method in action. The following video provides a very clear, step-by-step demonstration.

3. Visualising Ratios and Summarising the Method

The numerical method you just saw is fast and reliable. Sometimes, it also helps to visualise the problem. A "bar model" or "strip diagram" is a great way to do this, representing the ratio parts as blocks. The BBC Bitesize resource below introduces this visual method and provides a concise summary of the numerical steps.

4. Let's Practice

It's time to put your new skill to the test. Use the three-step method to solve the following problems.

Questions:

1. Two friends, Tama and Sophie, share a prize of $120 in the ratio 2:3. How much does each person get?
2. A recipe for a smoothie requires bananas, strawberries, and yoghurt in the ratio 1:3:2. If the total smoothie is 600 grams, what is the weight of each ingredient?
3. A farmer is planting a field with 350 trees. He plants kauri and rātā trees in the ratio 4:1. How many of each tree does he plant?

Solutions:

1. Share $120 in the ratio 2:3

   • Sum: Total parts = $2+3=5$ parts.
   • Divide: Value of one part = $\$120÷5=\$24$.
   • Multiply:
     • Tama's share (2 parts): $2\times \$24=\$48$.
     • Sophie's share (3 parts): $3\times \$24=\$72$.
   • (Check: $48 + $72 = $120)

2. Share 600 g in the ratio 1:3:2

   • Sum: Total parts = $1+3+2=6$ parts.
   • Divide: Value of one part = $600\text{ g}÷6=100\text{ g}$.
   • Multiply:
     • Bananas (1 part): $1\times 100\text{ g}=100\text{ g}$.
     • Strawberries (3 parts): $3\times 100\text{ g}=300\text{ g}$.
     • Yoghurt (2 parts): $2\times 100\text{ g}=200\text{ g}$.
   • (Check: 100 + 300 + 200 = 600 g)

3. Share 350 trees in the ratio 4:1

   • Sum: Total parts = $4+1=5$ parts.
   • Divide: Value of one part = $350÷5=70$ trees.
   • Multiply:
     • Kauri (4 parts): $4\times 70=280$ trees.
     • Rātā (1 part): $1\times 70=70$ trees.
   • (Check: 280 + 70 = 350 trees)

Key Takeaways

Well done! You've successfully learned how to share quantities using ratios. This is a fundamental skill in mathematics that appears in many different contexts.

• A ratio is a part-to-part comparison (e.g., 2:3).
• A fraction is a part-to-whole comparison (e.g., 2/5).
• To divide a quantity in a given ratio, use the three-step method: Sum the parts, Divide the total amount by the sum of the parts, and Multiply the result by each part of the ratio.

In our next lesson, we will shift gears slightly to look at relationships between different types of measurements. You will learn how to calculate one of speed, distance, or time when the other two are known.