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CAT 2020 Slot 2 — QA Question 4

BasicsEasy

John takes twice as much time as Jack to finish a job. Jack and Jim together take one-thirds of the time to finish the job than John takes working alone. Moreover, in order to finish the job, John takes three days more than that taken by three of them working together. In how many days will Jim finish the job working alone?

Answer & solution

Answer: 4

Solution

Easy

Work with efficiencies (work per day). Fix John's rate as the unit, derive Jack's and Jim's rates from the two ratio conditions, then use the "33 days more than all together" condition to find John's time — which equals Jim's time since their rates match.

1

Set efficiencies. Let John take time TT; "John takes twice as much time as Jack" means Jack is twice as fast.

eJohn=1T,eJack=2T\begin{aligned} &e_{\text{John}} = \tfrac{1}{T},\qquad e_{\text{Jack}} = \tfrac{2}{T} \end{aligned}
2

Find Jim's rate. Jack and Jim together take one-third of John's solo time, so their combined rate is three times John's.

eJack+eJim=3 eJohn=3T⇒ eJim=3T−2T=1T\begin{aligned} &e_{\text{Jack}} + e_{\text{Jim}} = 3\,e_{\text{John}} = \tfrac{3}{T}\\ &\Rightarrow\ e_{\text{Jim}} = \tfrac{3}{T} - \tfrac{2}{T} = \tfrac{1}{T} \end{aligned}

So Jim and John have the same rate (from step 1).

3

Use the "33 days more" condition. All three together have rate eJohn+eJack+eJim=4Te_{\text{John}}+e_{\text{Jack}}+e_{\text{Jim}} = \tfrac{4}{T}, so together they need T4\tfrac{T}{4} days. John (TT days) takes 33 more.

T=T4+3⇒ 3T4=3⇒ T=4 days\begin{aligned} &T = \frac{T}{4} + 3\\ &\Rightarrow\ \frac{3T}{4} = 3\\ &\Rightarrow\ T = 4\ \text{days} \end{aligned}
4

Jim's solo time. Since eJim=eJohn=1Te_{\text{Jim}} = e_{\text{John}} = \tfrac1T (step 2), Jim alone takes the same TT.

TJim=4 daysT_{\text{Jim}} = 4\ \text{days}
4 days4\ \text{days}
CAT 2020 Slot 2 QA Q4: John takes twice as much time as Jack to finish a job. Jack and Jim together take one-thirds of the time to fi — Solution | TheCATExam