Number · Ratios and Proportion · Lesson 3

Tables of Equivalent Ratios and Sharing in a Ratio

When Auntie Esi cooks jollof for a party of 50 instead of a family of 10, she does not guess: she scales up every ingredient in the same ratio. In this lesson you will build tables of equivalent ratios, link them to equivalent fractions, and use them to share money and materials fairly in a given ratio.

NaCCA B7.1.4.1.3 ⏱ about 50 min

By the end you will be able to

✔Generate equivalent ratios by multiplying or dividing both terms by the same number.
✔Write equivalent ratios as equivalent common fractions.
✔Make and complete tables of equivalent ratios for quantities that are proportional.
✔Share a quantity in a given ratio, including when only one person's share is known.

Key words

Equivalent ratios: Ratios that make the same comparison, e.g. 2 : 3, 4 : 6 and 6 : 9.
Equivalent fractions: Fractions with the same value, e.g. $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$ .
Proportional: Two quantities are proportional when they always stay in the same ratio, so one is always the same multiple of the other.
Share in a ratio: Divide a total into unequal parts so that the parts are in a given ratio.

Curriculum reference

Content standard: B7.1.4.1 Demonstrate an understanding of the concept of ratios and its relationship to fractions and use it to solve problems that involve rates, ratios, and proportional reasoning.

Indicator: Make tables of equivalent ratios (written as common fractions) relating quantities that are proportional.

Starter

Jollof for fifty guests

Auntie Esi's famous jollof uses 4 cups of rice to feed 10 people. Her nephew Yaw is getting married and 50 guests are coming. How many cups of rice does she need? If she uses too little, guests go home hungry; too much, and money is wasted. And after the wedding, Yaw and his two brothers must share the gift money fairly in the ratio of what each contributed. Both problems are solved with one powerful tool: a table of equivalent ratios.

1. Equivalent ratios are equivalent fractions

You already know that $\frac{2}{3} = \frac{4}{6} = \frac{8}{12}$ : multiplying the top and bottom of a fraction by the same number does not change its value. Ratios behave in exactly the same way. If you multiply or divide both terms of a ratio by the same (non-zero) number, you get an equivalent ratio.

▶ The shaded length never changes, so 2 : 3, 4 : 6 and 8 : 12 all describe the same comparison.

Ratio	× 2	× 3	× 4	As fractions
2 : 5	4 : 10	6 : 15	8 : 20	$\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20}$
3 : 4	6 : 8	9 : 12	12 : 16	$\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16}$

2. Making a table of equivalent ratios

When two quantities are proportional, a table of equivalent ratios lets you read off any amount you need. Auntie Esi's recipe is 4 cups of rice for 10 people. Each column of the table is the first column multiplied by the same number.

Every column has rice : people = 2 : 5, i.e. rice is $\frac{2}{5}$ of the number of people.
People	10	20	30	50
Cups of rice	4	8	12	20
Multiplier	× 1	× 2	× 3	× 5

So for 50 guests, Auntie Esi needs 20 cups of rice. Notice that $\frac{4}{10} = \frac{8}{20} = \frac{12}{30} = \frac{20}{50} = \frac{2}{5}$ . The fraction stays the same in every column, which is how you can test whether a table really shows proportional quantities.

1Write the given ratio as the first column.
2Multiply (or divide) BOTH quantities by 2, 3, 4, … to make new columns.
3Check: every column should simplify to the same fraction.

3. Sharing in a ratio

Kofi and Ama opened a provision shop. Kofi put in GH₵600 and Ama put in GH₵400, so they agreed to share the profit in the ratio 600 : 400 = 3 : 2. In the first month the profit was GH₵500. Think of the profit as 3 + 2 = 5 equal parts.

▶ Five equal boxes share the GH₵500. Kofi gets 3 boxes and Ama gets 2.

Sometimes you are told one person's share instead of the total. Kafui (36 years), Adoley (48 years) and Jantuah (24 years) shared money in the ratio of their ages. Jantuah received GH₵24,000. First make a table of equivalent ratios to find the simplest form:

Any column can be used; the simplest, 3 : 4 : 2, makes the arithmetic easiest.
Name	Age	÷ 2	÷ 4	÷ 12
Kafui	36	18	9	3
Adoley	48	24	12	4
Jantuah	24	12	6	2

▶ Jantuah's 2 boxes are worth GH₵24,000, so each box is GH₵12,000. Count all 9 boxes for the total.

Using fractions: Jantuah's share is $\frac{2}{9}$ of the total, so $\frac{2}{9}$ of the total = 24,000. Then $\frac{1}{9}$ of the total = 12,000 and the whole total = 9 × 12,000 = GH₵108,000.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Equivalent ratios as fractions

Write three ratios equivalent to 2 : 5 and express each as a common fraction.

Show the solution

Step 1Multiply both terms by 2: 4 : 10.
Step 2Multiply both terms by 3: 6 : 15.
Step 3Multiply both terms by 4: 8 : 20.
Step 4As fractions: $\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20}$ .

Answer: 4 : 10, 6 : 15, 8 : 20

Example 2

Scaling a jollof recipe

4 cups of rice feed 10 people. Make a table to find the rice needed for 20, 30 and 50 people.

Show the solution

Step 120 people = 10 × 2, so rice = 4 × 2 = 8 cups.
Step 230 people = 10 × 3, so rice = 4 × 3 = 12 cups.
Step 350 people = 10 × 5, so rice = 4 × 5 = 20 cups.
Step 4Check: $\frac{4}{10} = \frac{8}{20} = \frac{12}{30} = \frac{20}{50} = \frac{2}{5}$ ✓

Answer: 8, 12 and 20 cups

Example 3

Sharing provision-shop profit

Kofi and Ama share a profit of GH₵500 in the ratio 3 : 2. Find each share.

Show the solution

Step 1Total parts = 3 + 2 = 5.
Step 2One part = 500 ÷ 5 = GH₵100.
Step 3Kofi = 3 × 100 = GH₵300; Ama = 2 × 100 = GH₵200.
Step 4Check: 300 + 200 = 500 ✓

Answer: Kofi GH₵300, Ama GH₵200

Example 4

Sharing when one share is known

Kafui, Adoley and Jantuah shared money in the ratio of their ages, 36, 48 and 24 years. Jantuah received GH₵24,000. How much money did they share?

Show the solution

Step 1Simplify 36 : 48 : 24 by dividing by 12: 3 : 4 : 2. Total parts = 9.
Step 2Jantuah's 2 parts = GH₵24,000, so 1 part = GH₵12,000.
Step 3Total = 9 × 12,000 = GH₵108,000.
Step 4Check: Kafui 36,000 + Adoley 48,000 + Jantuah 24,000 = 108,000 ✓

Answer: GH₵108,000

Example 5

Mixing concrete

A mason mixes cement and sand in the ratio 1 : 4. He needs 30 headpans of mixture altogether. How many headpans of sand does he use?

Show the solution

Step 1Total parts = 1 + 4 = 5.
Step 2One part = 30 ÷ 5 = 6 headpans.
Step 3Sand = 4 × 6 = 24 headpans (and cement = 6 headpans).
Step 4Check: 6 + 24 = 30 and 6 : 24 = 1 : 4 ✓

Answer: 24 headpans of sand

Summary

Multiply or divide both terms by the same number to get an equivalent ratio; never add or subtract.
Equivalent ratios can be written as equivalent fractions: 2 : 3 = $\frac{2}{3} = \frac{4}{6}$ .
In a table of proportional quantities, every column simplifies to the same ratio.
To share in a ratio: add the parts, find one part, then multiply for each share.
If one share is known, divide it by its ratio number to find one part, then build the rest.

Jollof for fifty guests

1. Equivalent ratios are equivalent fractions

2. Making a table of equivalent ratios

3. Sharing in a ratio

Worked examples

Equivalent ratios as fractions

Scaling a jollof recipe

Sharing provision-shop profit

Sharing when one share is known

Mixing concrete

Summary

Practice: 15 questions