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Zorluk: OrtaRatios, Rates, and Proportions

A digital publishing company uses two high-speed printing presses, Press P and Press Q. Press P prints pages at a constant rate that is 40%40\% faster than the rate of Press Q. If Press P and Press Q work simultaneously at their respective constant rates, they can complete a printing job of 18,00018,000 pages in 55 hours. How many hours would it take Press Q, working alone at its constant rate, to complete a job of 15,00015,000 pages?

Cevap: 10 hours

Cevap

10
Working together, Press P and Press Q complete 18,00018,000 pages in 55 hours, which corresponds to a combined rate of 3,6003,600 pages per hour. Because Press P's rate is 1.41.4 times Press Q's rate, their combined rate is 2.42.4 times Press Q's rate. Dividing 3,6003,600 by 2.42.4 gives Press Q's individual rate of 1,5001,500 pages per hour. Finally, dividing 15,00015,000 pages by 1,5001,500 pages per hour yields 1010 hours.

Adım Adım Çözüm

1
Relate the rate of Press P to Press Q
rP=1.4rQr_P = 1.4 r_Q
Press P is 40% faster than Press Q, so its rate is 1+0.40=1.41 + 0.40 = 1.4 times the rate of Press Q.
2
Calculate the combined rate expression
rcombined=2.4rQr_{\text{combined}} = 2.4 r_Q
When working together, their rates add: rP+rQ=1.4rQ+rQ=2.4rQr_P + r_Q = 1.4 r_Q + r_Q = 2.4 r_Q.
3
Solve for the rate of Press Q (rQr_Q)
rQ=1,500r_Q = 1,500 pages per hour
Using Work=Rate×Time\text{Work} = \text{Rate} \times \text{Time}, we have 18,000=(2.4rQ)×5=12rQ18,000 = (2.4 r_Q) \times 5 = 12 r_Q. Dividing 18,00018,000 by 1212 yields rQ=1,500r_Q = 1,500.
4
Determine the time required for Press Q to complete 15,00015,000 pages
1010 hours
Dividing the target workload by Press Q's rate gives 15,000 pages1,500 pages/hour=10\frac{15,000\text{ pages}}{1,500\text{ pages/hour}} = 10 hours.

Anahtar Kavram

Combined Work Rates and Direct Proportions
Tahmini Süre:1m 30s
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Ratios, Rates, and Proportions

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