Chain Rule Questions and Answers

Answer: D

224 more workers are needed. The work is 560 × 7 = 3920 worker-months. In 5 months the workers needed are 3920 / 5 = 784. The extra workers are 784 - 560 = 224.

784 is the total required, not the extra, and 400 and 408 do not come from the inverse proportion.

Answer: D

320 prisoners were transferred. The provisions equal 800 x 15 = 12,000 prisoner-days. For 25 days they feed 12000/25 = 480 prisoners, so 800 - 480 = 320 were transferred. 200, 1120 and 480 are the other quantities that appear in the working but do not answer the question. Provisions can be treated as a fixed stock of prisoner-days. A stock that lasts 15 days for 800 people can last 25 days only for fewer people, in inverse proportion: 800 x 15 / 25 = 480.

Answer: A

Work in man-days, the quantity that stays fixed however the numbers change. After 3 days the provision left is what 500 men would eat in the remaining \( 27 - 3 = 24 \) days:

\[ \text{food remaining} = 500 \times 24 = 12{,}000 \text{ man-days} \]

The reinforcement brings the strength to \( 500 + 300 = 800 \), so

\[ \text{days} = \frac{12{,}000}{800} = 15 \]

Note the two traps: the 27 must be reduced to 24 before anything else, and the 300 is an addition to the garrison rather than the new total.

Answer: B

(2 x 14) men +(7 x 14) boys = (3 x 11) men + (8 x 11) boys

 =>5 men= 10 boys    => 1man= 2 boys

 Therefore, (2 men+ 7 boys) = (2 x 2 +7) boys = 11 boys

( 8 men + 6 boys) = (8 x 2 +6) boys = 22 boys.

 Let the required  number of days be x.

 More boys , Less days     (Indirect proportion)

 More work , More days    (Direct proportion)

 Boys22:11Work1 : 3⋮⋮ 14:x

 Therefore, (22 * 1 * x) = (11 * 3 * 14)

=> x = 21 

Hence, the required number of days = 21

Answer: D

Let the required length be x meters

 More men, More length built     (Direct proportion) 

Less days, Less length built      (Direct Proportion)

Men20:35Days6 : 3⋮⋮ 56 :x 

=> (20 x 6 x X)=(35 x 3 x 56) 

=> x = 49 

Hence, the required length is 49 m.

Answer: E

Let there be x men originally.
They were to complete the work in 180 days but as the number of persons is reduced to x - 30.
∴ Work takes 20 more days. So the equation is 180x = (x - 30)200 ⇒ 20x = 6000 ⇒ x = 300.

Answer: C

Count the food in man-days, which does not change when men leave. After 10 days there remain \( 45 - 10 = 35 \) days of provision for 150 men:

\[ \text{food remaining} = 150 \times 35 = 5{,}250 \text{ man-days} \]

With 25 men gone the strength is \( 150 - 25 = 125 \), so

\[ \text{days} = \frac{5{,}250}{125} = 42 \]

Fewer men must make the food last longer, so any answer below 35 is wrong on inspection - which disposes of 12 at once.

Answer: B

Find the daily rate first, then apply it to what is left. Completing \( \tfrac{5}{8} \) in 10 days means

\[ \text{rate} = \frac{5/8}{10} = \frac{1}{16} \text{ of the job per day} \]

The remaining fraction is \( 1 - \tfrac{5}{8} = \tfrac{3}{8} \), so

\[ \text{days} = \frac{3/8}{1/16} = \frac{3}{8} \times 16 = 6 \]

The question asks how many more days, so 6 is the answer; 16 would be the total time for the whole job from the start.

Answer: D

Reduce to a single weaver before scaling up. Four weavers making 4 mats in 4 days means

\[ 1 \text{ weaver} \rightarrow 1 \text{ mat in } 4 \text{ days} \]

Output is proportional to both the number of weavers and the time given, so

\[ \text{mats} = 8 \text{ weavers} \times \frac{8 \text{ days}}{4 \text{ days per mat}} = 8 \times 2 = 16 \]

The intuitive answer of 8 comes from doubling only one of the two factors. Both the workforce and the time double here, so the output goes up fourfold.

Answer: C

Read the rate per cow rather than for the herd. If 40 cows finish 40 bags in 40 days, then in those same 40 days

\[ \text{each cow eats } \frac{40 \text{ bags}}{40 \text{ cows}} = 1 \text{ bag} \]

so one cow takes 40 days for one bag.

The number of cows never enters the answer: each is eating alongside the others, not sharing a single bag. Dividing 40 by 40 to get 1 day is the classic slip, and it would mean the herd polished off 40 bags in a single day.