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07 Aug 2026, 04:49
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A textile workshop must produce a certain number of identical fabric rolls. The workshop has standard looms and industrial looms. Each standard loom produces fabric at the same constant rate, and each industrial loom produces fabric at the same constant rate. If 7 standard looms and 5 industrial looms, working together, can complete the job in 14 hours, and 6 industrial looms working alone can complete the job in 14 hours less time than 21 standard looms working alone, what is the ratio of the production rate of one industrial loom to the production rate of one standard loom?
A. 3
B. 5
C. 6
D. 7
E. 9
A. 3
B. 5
C. 6
D. 7
E. 9
07 Aug 2026, 04:49
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Official Solution:
A textile workshop must produce a certain number of identical fabric rolls. The workshop has standard looms and industrial looms. Each standard loom produces fabric at the same constant rate, and each industrial loom produces fabric at the same constant rate. If 7 standard looms and 5 industrial looms, working together, can complete the job in 14 hours, and 6 industrial looms working alone can complete the job in 14 hours less time than 21 standard looms working alone, what is the ratio of the production rate of one industrial loom to the production rate of one standard loom?
A. 3
B. 5
C. 6
D. 7
E. 9
Let \(x\) and \(y\) be the production rates of one standard loom and one industrial loom, respectively.
We need to find \(\frac{y}{x}\).
Since 7 standard looms and 5 industrial looms complete the job in 14 hours, the total work is:
\(\text{work} = \text{time} * \text{combined rate} = 14(7x + 5y)\)
So 21 standard looms working alone would take:
\(\text{time} = \frac{\text{work}}{\text{rate}} = \frac{14(7x + 5y)}{21x} = \frac{2(7x + 5y)}{3x}\) hours
And 6 industrial looms working alone would take:
\(\text{time} = \frac{\text{work}}{\text{rate}} = \frac{14(7x + 5y)}{6y} = \frac{7(7x + 5y)}{3y}\) hours
Since 6 industrial looms complete the job in 14 hours less time than 21 standard looms, we have:
\(\frac{2(7x + 5y)}{3x} - \frac{7(7x + 5y)}{3y} = 14\)
Multiply both sides by \(3xy\):
\(2y(7x + 5y) - 7x(7x + 5y) = 42xy\)
\(14xy + 10y^2 - 49x^2 - 35xy = 42xy\)
\(10y^2 - 63xy - 49x^2 = 0\)
Since we need to find \(\frac{y}{x}\), divide by \(x^2\):
\(10\left(\frac{y}{x}\right)^2 - 63\left(\frac{y}{x}\right) - 49 = 0\)
At this stage, it is easier to test the answer choices for \(\frac{y}{x}\) than to solve the quadratic. Only answer choice D works:
\(10(7)^2 - 63(7) - 49 = 490 - 441 - 49 = 0\)
Therefore, \(\frac{y}{x} = 7\), and thus the ratio of the production rate of one industrial loom to the production rate of one standard loom is 7.
Answer: D
A textile workshop must produce a certain number of identical fabric rolls. The workshop has standard looms and industrial looms. Each standard loom produces fabric at the same constant rate, and each industrial loom produces fabric at the same constant rate. If 7 standard looms and 5 industrial looms, working together, can complete the job in 14 hours, and 6 industrial looms working alone can complete the job in 14 hours less time than 21 standard looms working alone, what is the ratio of the production rate of one industrial loom to the production rate of one standard loom?
A. 3
B. 5
C. 6
D. 7
E. 9
Let \(x\) and \(y\) be the production rates of one standard loom and one industrial loom, respectively.
We need to find \(\frac{y}{x}\).
Since 7 standard looms and 5 industrial looms complete the job in 14 hours, the total work is:
\(\text{work} = \text{time} * \text{combined rate} = 14(7x + 5y)\)
So 21 standard looms working alone would take:
\(\text{time} = \frac{\text{work}}{\text{rate}} = \frac{14(7x + 5y)}{21x} = \frac{2(7x + 5y)}{3x}\) hours
And 6 industrial looms working alone would take:
\(\text{time} = \frac{\text{work}}{\text{rate}} = \frac{14(7x + 5y)}{6y} = \frac{7(7x + 5y)}{3y}\) hours
Since 6 industrial looms complete the job in 14 hours less time than 21 standard looms, we have:
\(\frac{2(7x + 5y)}{3x} - \frac{7(7x + 5y)}{3y} = 14\)
Multiply both sides by \(3xy\):
\(2y(7x + 5y) - 7x(7x + 5y) = 42xy\)
\(14xy + 10y^2 - 49x^2 - 35xy = 42xy\)
\(10y^2 - 63xy - 49x^2 = 0\)
Since we need to find \(\frac{y}{x}\), divide by \(x^2\):
\(10\left(\frac{y}{x}\right)^2 - 63\left(\frac{y}{x}\right) - 49 = 0\)
At this stage, it is easier to test the answer choices for \(\frac{y}{x}\) than to solve the quadratic. Only answer choice D works:
\(10(7)^2 - 63(7) - 49 = 490 - 441 - 49 = 0\)
Therefore, \(\frac{y}{x} = 7\), and thus the ratio of the production rate of one industrial loom to the production rate of one standard loom is 7.
Answer: D
