A textile mill uses two weaving looms, Loom P and Loom Q, to produce a fabric blend compos… — GRE General Test

A textile mill uses two weaving looms, Loom $P$ and Loom $Q$ , to produce a fabric blend composed of silk, wool, and cotton.

- Loom $P$ produces silk, wool, and cotton in the ratio $2 : 3 : 5$ by weight, operating at a constant output rate of $120\text{ kg per hour}$ .
- Loom $Q$ produces silk, wool, and cotton in the ratio $1 : 4 : 3$ by weight, operating at a constant output rate of $160\text{ kg per hour}$ .

If both looms operate simultaneously for 5 hours, what is the ratio of the total weight of wool produced to the total weight of cotton produced in the combined output?

$29 : 30$ Answer
B
$7 : 8$
C
$32 : 35$
D
$29 : 70$
E
$30 : 29$

Answer

The ratio of the total weight of wool produced to the total weight of cotton produced is

29 : 30

To find the overall ratio of wool to cotton, determine the actual mass of each material produced per hour by each machine. Loom

P

outputs

\frac{3}{10} \times 120 = 36\text{ kg}

of wool and

\frac{5}{10} \times 120 = 60\text{ kg}

of cotton per hour. Loom

Q

outputs

\frac{4}{8} \times 160 = 80\text{ kg}

of wool and

\frac{3}{8} \times 160 = 60\text{ kg}

of cotton per hour. Combining both looms yields

116\text{ kg}

of wool and

120\text{ kg}

of cotton per hour. Over 5 hours, the combined output is

580\text{ kg}

of wool and

600\text{ kg}

of cotton. The ratio of total wool to total cotton is

580 : 600

, which simplifies to

29 : 30

Step-by-Step Solution

Calculate the hourly production rates of wool and cotton for Loom

P

Loom

P

produces fabric at

120\text{ kg/hr}

with ratio parts

2 + 3 + 5 = 10

.
- Wool rate from

P = \frac{3}{10} \times 120 = 36\text{ kg/hr}

- Cotton rate from

P = \frac{5}{10} \times 120 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

P

's total hourly output.

Calculate the hourly production rates of wool and cotton for Loom

Q

Loom

Q

produces fabric at

160\text{ kg/hr}

with ratio parts

1 + 4 + 3 = 8

.
- Wool rate from

Q = \frac{4}{8} \times 160 = 80\text{ kg/hr}

- Cotton rate from

Q = \frac{3}{8} \times 160 = 60\text{ kg/hr}

Converting the ratio into component rates by multiplying each component's fraction by Loom

Q

's total hourly output.

Determine total quantities produced over the 5-hour period.

Total wool

= (36 + 80) \times 5 = 116 \times 5 = 580\text{ kg}

.
Total cotton

= (60 + 60) \times 5 = 120 \times 5 = 600\text{ kg}

Combining the hourly outputs from both machines and multiplying by the total duration of 5 hours.

Compute and simplify the ratio of total wool to total cotton.

\text{Ratio} = \frac{580}{600} = \frac{29}{30}

, which is expressed as

29 : 30$.

Dividing both terms of the ratio by their greatest common divisor, 20.

Key Concept

Combining weighted rates across multiple ratio-based sub-components

Estimated Time:2m 0s

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