TheCATExam
Sign in

CAT 2021 Slot 1 — QA Question 12

AlligationEasy

The strength of an indigo solution in percentage is equal to the amount of indigo in grams per 100 cc of water. Two 800 cc bottles are filled with indigo solutions of strengths 33% and 17%, respectively. A part of the solution from the first bottle is thrown away and replaced by an equal volume of the solution from the second bottle. If the strength of the indigo solution in the first bottle has now changed to 21% then the volume, in cc, of the solution left in the second bottle is

Answer & solution

Answer: 200

Solution

Easy

This is a replacement-mixture problem. When a volume xx is poured out of bottle 1 (strength 33%33\%) and replaced by the same volume from bottle 2 (strength 17%17\%), the resulting strength is a weighted average of the two. Use alligation to find what fraction was replaced, then subtract that volume from bottle 2.

1

Set up the mixture. Let xx cc be removed from bottle 1 (strength 33%33\%) and replaced by xx cc from bottle 2 (strength 17%17\%). The kept part of bottle 1 is (800−x)(800-x) cc at 33%33\%, and the added part is xx cc at 17%17\%. The new strength is 21%21\%.

2

Apply alligation. The mean strength 2121 splits the two components in the inverse ratio of their distances from the mean.

kept (33%)added (17%)=21−1733−21=412=13(alligation rule)⇒ 800−xx=13\begin{aligned} &\frac{\text{kept }(33\%)}{\text{added }(17\%)}=\frac{21-17}{33-21}=\frac{4}{12}=\frac{1}{3}\quad\text{(alligation rule)}\\ &\Rightarrow\ \frac{800-x}{x}=\frac{1}{3} \end{aligned}
3

Solve for the replaced volume xx.

3(800−x)=x(cross-multiply)⇒ 2400−3x=x⇒ 4x=2400⇒ x=600 cc\begin{aligned} &3(800-x)=x\quad\text{(cross-multiply)}\\ &\Rightarrow\ 2400-3x=x\\ &\Rightarrow\ 4x=2400\\ &\Rightarrow\ x=600\ \text{cc} \end{aligned}
4

Find what is left in bottle 2. Bottle 2 started with 800800 cc and gave away x=600x=600 cc.

800−600=200 cc\begin{aligned} &800-600=200\ \text{cc} \end{aligned}
200 cc200\ \text{cc}

Strength moved from 33%33\% to 21%21\%, i.e. 33−2133−17=1216=34\tfrac{33-21}{33-17}=\tfrac{12}{16}=\tfrac34 of the way toward 17%17\%. So 34\tfrac34 of the 800800 cc was replaced: 600600 cc removed, leaving 800−600=200800-600=200 cc in bottle 2.

CAT 2021 Slot 1 QA Q12: The strength of an indigo solution in percentage is equal to the amount of indigo in grams per 100 cc of water — Solution | TheCATExam