Question

A dishonest shopkeeper professes to sell pulses at his cost price but uses a false weight of $950gm$ for Kilogram. Thus his gain percent is a $a\dfrac{b}{c}\% $. Find the value of $a + b + c$.

Answer 1

Hint: We are given that the shopkeeper sells pulses at his cost price but uses a false weight of $950gm$ for Kilogram. The gain percent is $a\dfrac{b}{c}\% $. Here, we need to find the profit/gain percent which is given in the form of $a\dfrac{b}{c}\% $ and then we will get value of $a$, $b$ and $c$. We can find the profit percent using the formula given below.
Formula used: $\Pr ofit = \left( {\dfrac{{SP - CP}}{{CP}} \times 100} \right)\% $

Complete step-by-step solution:
We are given that a dishonest shopkeeper sells pulses at his cost price but uses a false weight of $950gm$ for $1kg$.
Given: Selling weight is $1kg$ or $1000g$ and cost weight is $950gm$.
We have to find the profit/gain %.
$\Pr ofit = \left( {\dfrac{{SP - CP}}{{CP}} \times 100} \right)\% $ , where $SP$ is the selling weight, $CP$ is the cost weight.
Substitute value of Selling weight and cost weight
\[ \Rightarrow \Pr ofit = \left( {\dfrac{{1000 - 950}}{{950}} \times 100} \right)\% \]
\[ \Rightarrow \Pr ofit = \left( {\dfrac{{50}}{{950}} \times 100} \right)\% \]
On multiplying, we get
\[ \Rightarrow \Pr ofit = \left( {\dfrac{{5000}}{{950}}} \right)\% \]
On simplifying, we get
\[ \Rightarrow \Pr ofit = \left( {\dfrac{{100}}{{19}}} \right)\% \]
Therefore, his gain percent is $\dfrac{{100}}{{19}}\% $
As already mentioned in the question that gain percent is $a\dfrac{b}{c}\% $, so we will convert $\dfrac{{100}}{{19}}\% $ into mixed fraction.
$ \Rightarrow \dfrac{{100}}{{19}}\% = 5\dfrac{5}{{19}}\% $
On comparing $a\dfrac{b}{c}\% $ and $5\dfrac{5}{{19}}\% $, we get
$a = 5$, $b = 5$ and $c = 19$.
We have to find the value of $a + b + c$.
On substituting value of $a = 5$, $b = 5$ and $c = 19$, we get
$ = 5 + 5 + 19$
$ = 29$
Hence, $a + b + c = 29$.

Note: Here, cost weight is the real weight (true weight), and selling weight is the false weight in which the shopkeeper pretends it as true. To convert an improper fraction into a mixed fraction, divide the numerator by denominator and obtain quotient and remainder. We write mixed fraction as: $Quotient\dfrac{{\operatorname{Re} mainder}}{{Deno\min ator}}$