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CAT 2018 Slot 2 — QA Question 4

Pipes & CisternsEasy

A water tank has inlets of two types A and B. All inlets of type A when open, bring in water at the same rate. All inlets of type B, when open, bring in water at the same rate. The empty tank is completely filled in 30 minutes if 10 inlets of type A and 45 inlets of type B are open, and in 1 hour if 8 inlets of type A and 18 inlets of type B are open. In how many minutes will the empty tank get completely filled if 7 inlets of type A and 27 inlets of type B are open?

Answer & solution

Answer: 48

Solution

Easy

Let one type-A inlet fill at rate aa and one type-B inlet at rate bb (units/min). The two given fill scenarios both equal the same full tank, so equate them to relate aa and bb; then compute the time for the third configuration.

1

Equate the two full-tank scenarios. 3030 min with (10a+45b)(10a+45b) and 6060 min with (8a+18b)(8a+18b) both fill the same tank.

30(10a+45b)=60(8a+18b)⇒ 300a+1350b=480a+1080b⇒ 270b=180a⇒ a=1.5 b\begin{aligned} &30(10a+45b) = 60(8a+18b)\\ &\Rightarrow\ 300a+1350b = 480a+1080b\\ &\Rightarrow\ 270b = 180a\\ &\Rightarrow\ a = 1.5\,b \end{aligned}
2

Tank capacity in terms of bb. Use the first scenario.

V=30(10a+45b)=300a+1350b⇒ V=300(1.5b)+1350b⇒ V=1800 b\begin{aligned} &V = 30(10a+45b) = 300a+1350b\\ &\Rightarrow\ V = 300(1.5b)+1350b\\ &\Rightarrow\ V = 1800\,b \end{aligned}
3

Rate of the third configuration. 77 type-A and 2727 type-B inlets:

r=7a+27b=7(1.5b)+27b⇒ r=10.5b+27b=37.5 b units/min\begin{aligned} &r = 7a+27b = 7(1.5b)+27b\\ &\Rightarrow\ r = 10.5b+27b = 37.5\,b\ \text{units/min} \end{aligned}
4

Time to fill.

t=Vr=1800b37.5b⇒ t=48 minutes\begin{aligned} &t = \frac{V}{r} = \frac{1800b}{37.5b}\\ &\Rightarrow\ t = 48\ \text{minutes} \end{aligned}
t=48 minutest = 48\ \text{minutes}
CAT 2018 Slot 2 QA Q4: A water tank has inlets of two types A and B. All inlets of type A when open, bring in water at the same rate. — Solution | TheCATExam