Practice Exercises

Two poles, one $78 \mathrm{m}$ in height above the ground and the other $91 \mathrm{m}$ in height above the ground, are at some distance from each other. Two strings are tied, one from the top of one pole to the bottom of the other and the other from the top of the second pole to the bottom of the first. What is the height above the ground at which the string meets?

(Note: You may wonder why this question, which is apparently using the concepts of geometry, is placed in this chapter. Look at the solution to find the traces of the similarity in the formation of this problem and the time and work concept.)

Two Pipes A and B can fill a cistern in $12\frac{1}{2}$ and $15 \mathrm{h}$, respectively. If the pipes can be opened or closed only after every $30 \min$ (i.e., at$6.00, 6.30$,etc.) and the tank is to be filled by using both the pipes without any overflow then,

To make an article, it takes $40 \mathrm{h}$ for a workman who is paid $₹ 1.80$ per hour. $20\%$ the material of is wasted in the course of working, which costs ₹ $22.5$ per kilogram. At what price, the article must be sold so as to yield a profit of $33.33\%$, if its weight was measured to be $8 \mathrm{kg}$?

Anand got an order from a boutique for $480$shirts. He brought $12$sewing machines and appointed some expert tailors to do the job. However, many did not reportto duty. As a result, each of those who reported had to stitch $32$more shirts than was originally planned by Anand, with equal distribution of work. How many tailors had been appointed earlier and how many had not reported to work?

The rate of flow of water (in litre per min) of three pipes are $2$, $\mathrm{N}$and $3$, where $2 < \mathrm{N} < 3$. The lowest and the highest flow rates are both decreased by a certain quantity x, while the intermediate rate is left unchanged. If the reciprocals of the three flow rates, in the order given above, are in arithmetic progression both before and after the change, then what is the quantity x (in litre per min)? (Negative flow rates indicate that the pipes act as emptying pipes instead of filling pipes.

The daily work of $2$men is equal to that of $3$women or that of $4$youngsters. By employing $14$men, $12$women, and $12$youngsters a certain work can be finished in $24$days. If it is required to finish it in $14$days and as an additional labour, only men are available, how many of them will be required?

A, B, and Care assigned a piece of job that they can complete by working together in $15$ days. Their efficiencies (measured in terms of rate of doing job) are in the ratio of $1:2:3$. After one-third of the job is completed, one of them has to be withdrawn due to budget constraint. Their wages per day are in the ratio of $3 : 5 : 6$. The number of days in which the remaining two persons can finish the job (at optimal cost) is:

Labour allocation is a very important process. A particular weaving section has $20$ looms and with five labourers, loom efficiency is $75\%$. The production of a loom at $100\%$ efficiency is $10 \mathrm{m}/\mathrm{h}$. Salary of a labourer is $₹11,000$per month. I removed one labourer due to which the efficiency came down to $70\%$. How much do I gain or loose due to this action? (Assume that the profit on $1 \mathrm{m}$ cloth is $₹ 4$ and the looms are working for $30$days in a month and $10$hours per day.)