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True - Discount
1.A man purchased a cow for Rs. 3000 and sold it the same day for Rs. 3600, allowing the buyer a credit of 2 years. If the rate of interest be 10% per annum, then the man has a gain of:

Explanation:

Solution:
$\begin{array}{rl}& \text{C}\text{.P}\text{.}=Rs.\phantom{\rule{thinmathspace}{0ex}}3000.\\ & \text{S}\text{.P}\text{.}=Rs.\phantom{\rule{thinmathspace}{0ex}}\left[\frac{3600×100}{100+\left(10×2\right)}\right]\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}3000.\\ & \text{Gain}=0\mathrm{%}.\end{array}$

2.The true discount on Rs. 2562 due 4 months hence is Rs. 122. The rate percent is:

Explanation:

Solution:
$\begin{array}{rl}& \text{P}\text{.W}\text{.}=Rs.\left(2562-122\right)\\ & =Rs.2440\\ & \therefore \text{S}\text{.I}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{2440}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{4}\phantom{\rule{thinmathspace}{0ex}}\text{months}\phantom{\rule{thinmathspace}{0ex}}\text{is}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{122}\text{.}\\ & \therefore \text{Rate}=\left[\frac{100×122}{2440×\frac{1}{3}}\right]\mathrm{%}=15\mathrm{%}\end{array}$

3.A trader owes a merchant Rs. 10,028 due 1 year hence. The trader wants to settle the account after 3 months. If the rate of interest 12% per annum, how much cash should he pay?

Explanation:

Solution:

4.If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount allowed on the same sum due at the end of double the time is:

Explanation:

Solution:
S.I. on Rs. (110 - 10) for a certain time = Rs. 10.
S.I. on Rs. 100 for double the time = Rs. 20.
T.D. on Rs. 120 = Rs. (120 - 100) = Rs. 20.
$\begin{array}{rl}& \text{T}\text{.D}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{110}\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}\left(\frac{20}{120}×110\right)\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}18.33\end{array}$

5.Goods were bought for Rs. 600 and sold the same for Rs. 688.50 at a credit of 9 months and thus gaining 2% The rate of interest per annum is:

Explanation:

Solution:
$\begin{array}{rl}& \text{S}\text{.P}\text{.}=102\mathrm{%}\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}Rs.\phantom{\rule{thinmathspace}{0ex}}600\\ & =\left(\frac{102}{100}×600\right)\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}612.\\ & \text{Now,}\phantom{\rule{thinmathspace}{0ex}}\text{P}\text{.W}\text{. = Rs}\text{.612}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\text{sum}\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}688.50.\\ & \therefore \text{T}\text{.D}\text{.}=Rs.\phantom{\rule{thinmathspace}{0ex}}\left(688.50-612\right)\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}76.50.\\ & \text{Thus,}\phantom{\rule{thinmathspace}{0ex}}\text{S}\text{.I}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{612}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{9}\phantom{\rule{thinmathspace}{0ex}}\text{months}\phantom{\rule{thinmathspace}{0ex}}\text{is}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{76}\text{.50}\text{.}\\ & \therefore \text{Rate}=\left(\frac{100×76.50}{612×\frac{3}{4}}\right)\mathrm{%}\\ & =16\frac{2}{3}\mathrm{%}\phantom{\rule{thinmathspace}{0ex}}\end{array}$

6.The true discount on a bill due 9 months hence at 16% per annum is Rs. 189. The amount of the bill is:

Explanation:

Solution:
$\begin{array}{rl}& \text{Let}\phantom{\rule{thinmathspace}{0ex}}\text{P}\text{.W}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{be}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{x}\text{.}\\ & \text{Then,}\phantom{\rule{thinmathspace}{0ex}}\text{S}\text{.I}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{x}\phantom{\rule{thinmathspace}{0ex}}\text{at}\phantom{\rule{thinmathspace}{0ex}}\text{16}\mathrm{%}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{9}\phantom{\rule{thinmathspace}{0ex}}\text{months}\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}189.\\ & \therefore x×16×\frac{9}{12}×\frac{1}{100}\\ & =189\phantom{\rule{thinmathspace}{0ex}}or\phantom{\rule{thinmathspace}{0ex}}x=1575.\\ & \therefore P.W.=Rs.\phantom{\rule{thinmathspace}{0ex}}1575.\\ & \therefore \text{Sum}\phantom{\rule{thinmathspace}{0ex}}\text{due}=P.W.+T.D.\\ & =Rs.\left(1575+189\right)\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}1764.\end{array}$

7.A man buys a watch for Rs. 1950 in cash and sells it for Rs. 2200 at a credit of 1 year. If the rate of interest is 10% per annum, the man:

Explanation:

Solution:
$\begin{array}{rl}& \text{S}\text{.P}\text{. = P}\text{.W}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{of}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{2200}\phantom{\rule{thinmathspace}{0ex}}\text{due}\phantom{\rule{thinmathspace}{0ex}}\text{1}\phantom{\rule{thinmathspace}{0ex}}\text{year}\phantom{\rule{thinmathspace}{0ex}}\text{hence}\\ & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=Rs.\phantom{\rule{thinmathspace}{0ex}}\left[\frac{2200×100}{100+\left(10×1\right)}\right]\\ & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=Rs.\phantom{\rule{thinmathspace}{0ex}}2000.\\ & \therefore \text{Gain}=Rs.\left(2000-1950\right)\\ & \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=Rs.\phantom{\rule{thinmathspace}{0ex}}50.\end{array}$

8.The present worth of Rs. 2310 due 21/2 years hence, the rate of interest being 15% per annum, is:

Explanation:

Solution:
$\begin{array}{rl}& P.W.=Rs.\phantom{\rule{thinmathspace}{0ex}}\left[\frac{100×2310}{100+\left(15×\frac{5}{2}\right)}\right]\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}1680.\end{array}$

9.Rs. 20 is the true discount on Rs. 260 due after a certain time. What will be the true discount on the same sum due after half of the former time, the rate of interest being the same?

Explanation:

Solution:
$\begin{array}{rl}& \text{S}\text{.I}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\left(\text{260 - 20}\right)\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{a}\phantom{\rule{thinmathspace}{0ex}}\text{given}\phantom{\rule{thinmathspace}{0ex}}\text{time}\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}20.\\ & \text{S}\text{.I}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{on}\phantom{\rule{thinmathspace}{0ex}}\text{Rs}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{240}\phantom{\rule{thinmathspace}{0ex}}\text{for}\phantom{\rule{thinmathspace}{0ex}}\text{half}\phantom{\rule{thinmathspace}{0ex}}\text{time}\\ & =Rs.10.\\ & T.D.\phantom{\rule{thinmathspace}{0ex}}on\phantom{\rule{thinmathspace}{0ex}}Rs.\phantom{\rule{thinmathspace}{0ex}}250=Rs.10.\\ & \therefore T.D.\phantom{\rule{thinmathspace}{0ex}}on\phantom{\rule{thinmathspace}{0ex}}Rs.\phantom{\rule{thinmathspace}{0ex}}260\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}\left(\frac{10}{250}×260\right)\\ & =Rs.\phantom{\rule{thinmathspace}{0ex}}10.40\end{array}$

10.The interest on Rs. 750 for 2 years is the same as the true discount on Rs. 960 due 2 years hence. If the rate of interest is the same in both cases, it is:

$\begin{array}{rl}& \therefore \text{Rate}=\left(\frac{100×210}{750×2}\right)\mathrm{%}\\ & =14\mathrm{%}\end{array}$