Making the Units Equal

Here is a statement that looks scary but hides an easy ratio: "Two-fifths of Ali's money is equal to one-third of Ben's money." Two different people, two different fractions — how can you compare them? The trick is to make the numerators equal, so that one "chunk" is the same size on both bars. The moment the chunks match, the whole thing becomes a clean ratio you already know how to solve.

Parents: let your child predict the ratio before revealing. The "Method tip" boxes name the move a marker rewards — making the numerators equal so one chunk is the same size on both bars.

By the end you'll be able to turn an equal-fraction statement into a ratio and find one unit. Let's match the chunks.

Make the numerators equal

A fraction like two-fifths means "2 chunks out of 5". If two-fifths of Ali equals one-third of Ben, then "2 of Ali's chunks" equals "1 of Ben's chunks" — but those chunks are different sizes, so you cannot compare yet.

Fix it by making the numerators the same. Rewrite one-third as two-sixths (multiply top and bottom by 2). Now both fractions have a numerator of 2: two-fifths of Ali equals two-sixths of Ben. So 2 equal chunks of Ali equal 2 equal chunks of Ben — which means one chunk is the same size on both. Ali is 5 of those chunks and Ben is 6.

From equal chunks to one unit

Once the chunks are equal, Ali is 5 units and Ben is 6 units of the same size. Now it is just a comparison model. Suppose Ben has $30 more than Ali. The difference is 6 − 5 = 1 unit = $30. Step one unit until the gap between the bars is $30.

🔬

Match the chunks, then find one unit

🤔 Predict first: After matching the chunks, what is the ratio of Ali to Ben?

5 : 62 : 3

With 1 unit = $30, Ali has 5 × 30 = $150 and Ben has 6 × 30 = $180. Check the original statement: two-fifths of 150 is $60, and one-third of 180 is also $60. They match.

Why the denominators become the units

It feels like magic, but it is simple. When both fractions take the same number of chunks (the equal numerators), one chunk is one shared unit. The fraction two-fifths means Ali is made of 5 of those chunks; two-sixths means Ben is made of 6. So after matching numerators, the denominators tell you the units — Ali : Ben = 5 : 6.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. To compare 1/2 of A with 1/3 of B by making the numerators equal, you do…

   Nothing, they already matchRewrite as 2/4 of A and 2/6 of BAdd the fractions

2. After matching numerators you get 2/4 of A equals 2/6 of B. What is the ratio A : B?

   4 : 66 : 42 : 2

3. 2/5 of Ali equals 1/3 of Ben. What is the ratio Ali : Ben?

   5 : 62 : 15 : 3

4. Ali : Ben = 5 : 6 and Ben has $40 more than Ali. What is one unit?

   $40$8$20

5. 3/4 of P equals 3/7 of Q. What is the ratio P : Q?

   4 : 73 : 37 : 4

6. 2/3 of Mara equals 1/2 of Nita. If they have $350 altogether, how much does Mara have?

   $150$200$175

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