At a manufacturing facility, the ratio of the daily output of Machine X to Machine Y is 4 … — GRE General Test

At a manufacturing facility, the ratio of the daily output of Machine $X$ to Machine $Y$ is $4 : 5$ , and the ratio of the daily output of Machine $Y$ to Machine $Z$ is $3 : 2$ . If the daily output of Machine $X$ is decreased by $10\%$ and the daily output of Machine $Z$ is increased by $25\%$ , what is the new ratio of the daily output of Machine $X$ to the daily output of Machine $Z$ ?

A
$6 : 5$
B
$27 : 25$
$108 : 125$ Answer
D
$9 : 10$
E
$48 : 125$

Answer

The new ratio of the daily output of Machine

X

to Machine

Z

108 : 125

The unified ratio of the outputs of the three machines is

X : Y : Z = 12 : 15 : 10

. Decreasing Machine

X

's output by

10\%

changes its relative units to

12 \times 0.90 = 10.8

. Increasing Machine

Z

's output by

25\%

changes its relative units to

10 \times 1.25 = 12.5

. Comparing the updated values yields

10.8 : 12.5

, which simplifies to

108 : 125

Step-by-Step Solution

Unify the two separate ratios into a single three-part ratio

X : Y : Z

Since

X : Y = 4 : 5

and

Y : Z = 3 : 2

, multiply

X : Y

3

to get

12 : 15

and multiply

Y : Z

5

to get

15 : 10

. Thus,

X : Y : Z = 12 : 15 : 10

Machine

Y

is the common element linking both ratios, so its ratio component must be equalized.

Calculate the updated values for

X

and

Z

after applying their respective percentage changes.

Machine

X_{new} = 12 \times (1 - 0.10) = 12 \times 0.90 = 10.8

. Machine

Z_{new} = 10 \times (1 + 0.25) = 10 \times 1.25 = 12.5

Applying a

10\%

decrease scales a quantity by

0.90

, while a

25\%

increase scales it by

1.25

Form the ratio

X_{new} : Z_{new}

and convert to lowest integer terms.

10.8 : 12.5 = \frac{10.8}{12.5} = \frac{108}{125}

, which gives the ratio

108 : 125

Multiplying both terms by

10

eliminates decimals, yielding the coprime integer ratio

108 : 125

Key Concept

Three-Part Ratio Unification and Relative Percentage Modification

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

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