Three pieces of fabric have a combined length of 120m. After cutting 35 of the first piece, 23 of the second piece, and 12 of the third piece, the remaining lengths of all three pieces are equal. What were the original lengths of each piece?

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Fractions/Ratios/Decimals

Three pieces of fabric have a combined length of 120m. After cutting $\frac{3}{5}$ of the first piece, $\frac{2}{3}$ of the second piece, and $\frac{1}{2}$ of the third piece, the remaining lengths of all three pieces are equal. What were the original lengths of each piece?

Answer Choices

Correct answer marked below

40m; 30m; 50m

40m; 48m; 32m

Correct

30m; 40m; 32m

50m; 30m; 40m

50m; 40m; 32m

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Explanation

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Official Explanation

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Three Pieces of Fabric Length Problem

Understanding the Problem

We have three pieces of fabric with original lengths:

First piece: $x$ meters

Second piece: $y$ meters

Third piece: $z$ meters

Their combined length is 120 meters:

$x + y + z = 120$

Fractions cut from each piece:

First piece: $\frac{3}{5}$ cut → remaining: $\frac{2}{5}x$

Second piece: $\frac{2}{3}$ cut → remaining: $\frac{1}{3}y$

Third piece: $\frac{1}{2}$ cut → remaining: $\frac{1}{2}z$

After cutting, remaining lengths are equal:

$\frac{2}{5}x = \frac{1}{3}y = \frac{1}{2}z$

Setting Up Equations

Let equal remaining length be $k$ :

$\frac{2}{5}x = k$ → $x = \frac{5}{2}k$

$\frac{1}{3}y = k$ → $y = 3k$

$\frac{1}{2}z = k$ → $z = 2k$

Substitute into total length equation:

…

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Comments

AlkaJan 27, 2025

This is exactly what I needed to practice

joaquinSep 30, 2024

Great question!

What this question tests

What it tests

Your ability to convert between fractions, ratios, and decimals and to reason proportionally.

Common trap

Cross-multiplying incorrectly or mixing up the ratio parts — a part:part ratio is not a fraction of the whole.

Three pieces of fabric have a combined length of 120m. After cutting 35\frac{3}{5}53​ of the first piece, 23\frac{2}{3}32​ of the second piece, and 12\frac{1}{2}21​ of the third piece, the remaining lengths of all three pieces are equal. What were the original lengths of each piece?