Making Units Equal

Two fractions can look unrelated even when the pieces they describe are the same size. If $\frac{2}{5}$ of a red ribbon roll equals $\frac{3}{8}$ of a blue ribbon roll, the fractions are not equal as numbers. Their actual portions are equal. Make those portions line up and the two whole lengths become a ratio.

Parents: let your child predict which numbers should match before revealing the common portion, then ask them to check both original fractions at the end.

By the end you will be able to recognise equal fractional portions, make their numerators match, read the resulting ratio, and solve for the requested amount.

Know when the portions are equal

Use this method when a question says that a fraction of one whole has the same amount as a fraction of another whole. The two wholes may be ribbons, money or collections of objects. A statement such as “$\frac{2}{5}$ of red equals $\frac{3}{8}$ of blue” is about two equal portions from two different bars.

The numerator counts the pieces in the known portion. The denominator tells how many of those pieces make the whole. First match the number of pieces in the two known portions; only then can you compare the wholes.

Make the numerators match

Here is the full question:

$\frac{2}{5}$ of a red ribbon roll has the same length as $\frac{3}{8}$ of a blue ribbon roll. The two rolls are 186 centimetres long altogether. Find the length of each roll.

The equality is between $\frac{2}{5}$ of red and $\frac{3}{8}$ of blue. Make the highlighted portions both 6 small units, the lowest common multiple of 2 and 3. For red, two original parts become six common units, so each red part is three common units and the whole is fifteen. For blue, three original parts become six common units, so each blue part is two common units and the whole is sixteen.

Match the equal portions

$$\frac{2}{5}\text{ of red} = \frac{3}{8}\text{ of blue}$$

$$2a = 6u,\quad 5a = 15u$$

$$3b = 6u,\quad 8b = 16u$$

Now both highlighted portions contain 6 equal units. The individual common unit has the same length on both bars. The red whole has 15 units and the blue whole has 16 units, so the ratio is 15 : 16.

Turn the ratio into one unit

After alignment, red has 15 units and blue has 16 units. Together the rolls are 186 cm, so 31 common units make 186 cm.

Find one unit

$$\text{Red : Blue} = 15 : 16$$

$$15u + 16u = 31u$$

$$1u = 186 \div 31 = 6\text{ cm}$$

$$\text{Red} = 15 \times 6 = 90\text{ cm}$$

$$\text{Blue} = 16 \times 6 = 96\text{ cm}$$

Use the adjustable bar trainer to set one common unit from the total. Predict the ratio first, then move the value until 31 units total 186 cm.

🔬

Equal portions: find one common unit

🤔 Predict first: After matching the portions, what is red : blue?

15 : 165 : 616 : 15

Check both original conditions. $\frac{2}{5}$ of 90 cm is 36 cm. $\frac{3}{8}$ of 96 cm is also 36 cm. The portions match, and the totals add to 186 cm.

Check the story

$$\frac{2}{5} \times 90 = 36\text{ cm}$$

$$\frac{3}{8} \times 96 = 36\text{ cm}$$

$$90 + 96 = 186\text{ cm}$$

Why denominators become the units

$\frac{2}{5}$ means the red whole is made of 5 equal pieces, while $\frac{3}{8}$ means the blue whole is made of 8 equal pieces. After the equal portions are rescaled to 6 common units, red has 15 units and blue has 16 units of the same size. The denominator counts all the original pieces; the rescaled total gives the comparable ratio.

Watch out — easily mixed up

Quick recap

Graduated practice

Open each solution only after you have written the equal portions and the resulting ratio.

1. Strings and a total

$\frac{3}{4}$ of Alia's string equals $\frac{2}{3}$ of Ben's string. Together the strings are 170 cm long. Find each length.

Show solution

Make both portions 6 units: $\frac{3}{4}$ becomes $\frac{6}{8}$, and $\frac{2}{3}$ becomes $\frac{6}{9}$. The whole lengths are 8 and 9 units.

Practice 1

$$\frac{3}{4} = \frac{6}{8}$$

$$\frac{2}{3} = \frac{6}{9}$$

$$8u + 9u = 170\text{ cm}$$

$$1u = 170 \div 17 = 10\text{ cm}$$

$$\text{Alia} = 8 \times 10 = 80\text{ cm}$$

$$\text{Ben} = 9 \times 10 = 90\text{ cm}$$

The equal portions are both 60 cm, and 80 + 90 = 170 cm.

2. Cards and a difference

$\frac{1}{3}$ of a box of rose cards equals $\frac{2}{5}$ of a box of tulip cards. There are 18 more rose cards than tulip cards. How many cards are in each box?

Show solution

Match the portions at 2 units. The rose box is 6 units and the tulip box is 5 units, so the one-unit difference is 18 cards.

Practice 2

$$\frac{1}{3} = \frac{2}{6}$$

$$\frac{2}{5}\text{ stays as }\frac{2}{5}$$

$$6u - 5u = 18\text{ cards}$$

$$1u = 18\text{ cards}$$

$$\text{Rose} = 6 \times 18 = 108$$

$$\text{Tulip} = 5 \times 18 = 90$$

$\frac{1}{3}$ of 108 and $\frac{2}{5}$ of 90 are both 36 cards.

3. Beads and a non-unit gap

$\frac{2}{7}$ of Cora's beads equals $\frac{4}{5}$ of Dan's beads. Cora has 81 more beads than Dan. Find both amounts.

Show solution

The numerators already match at 4 after changing $\frac{2}{7}$ to $\frac{4}{14}$. Cora has 14 units and Dan has 5 units. Their difference is 9 units.

Practice 3

$$\frac{2}{7} = \frac{4}{14}$$

$$14u - 5u = 81$$

$$1u = 81 \div 9 = 9$$

$$\text{Cora} = 14 \times 9 = 126$$

$$\text{Dan} = 5 \times 9 = 45$$

Both named portions are 36 beads: $\frac{2}{7}$ of 126 and $\frac{4}{5}$ of 45.

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. $\frac{2}{5}$ of red equals $\frac{3}{8}$ of blue. What should be aligned first?

   The actual portionsThe two whole lengthsThe denominators only

2. What common numerator aligns the 2 red parts and 3 blue parts?

   236

3. After alignment, what is the red : blue whole ratio?

   15 : 165 : 816 : 15

4. The two ribbon rolls total 186 cm across 31 common units. What is one unit?

   6 cm31 cm186 cm

5. $\frac{3}{4}$ of a ribbon equals $\frac{2}{3}$ of another. Together they are 170 cm. Which pair is correct?

   80 cm and 90 cm60 cm and 110 cm85 cm and 85 cm

6. $\frac{1}{3}$ of rose cards equals $\frac{2}{5}$ of tulip cards. Rose has 18 more cards. What are the counts?

   108 and 9036 and 5490 and 108

7. Why is making denominators equal by itself not enough when the two wholes are different?

   Equal denominators do not guarantee equal-sized piecesDenominators are never usefulFractions cannot be rewritten

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