Question

Latha bought a pencil and a pen for seven and a half rupees. She gave her Rs 10/-. The shopkeeper gave back the money in half and quarter rupees. What are the coins she got?

Answer 1

Hint: First we find how the amount the shopkeeper has to return to Latha. We use half a rupee is 50 paisa and a quarter rupee is 25 paisa. We try to add up the multiples of 50 and 25 so their sum of will be the same as the amount the shopkeeper returned.

Complete step-by-step solution:
We have from the question that Latha has bought a pen and a pencil with seven and a half rupees. We know that 1 rupee is 100 paisa. So half rupee will be $100\div 2=50$ paisa. If we write the price in numbers it will be 7 plus 50 paise, which is 7.50 rupees. Latha gave the shopkeeper 10 rupees. So the shopkeeper will return $10-7.50=2.50$ rupees. So the shopkeeper returned Rs.$2.50=2.50\times 100=250$ paisa.\[\]
The shopkeeper gave back the money in half rupee coins and quarter rupee coins. We know that a half rupee is 50 paisa. A quarter is one-fourth of a rupee. So one quarter is $100\times \dfrac{1}{4}=25$ paisa. We have to find how many coins of 50 paise and 25 paise are in 250 paise. \[\]
The multiples of 50 less than or equal to 250 are 50, 100, 150,200,250. The multiples of 25 less than 25 are 25,50,75,100,125,150,175,200,225,250. There have to be two types of coins in our selection. . If we select an odd number of 25 paisa coins we cannot add up the 50 paisa coins or half rupee coins to add up to 250. So we can take only an even number of 25 paisa or quarter rupee coins.\[\]
If there are 2 quarter rupee coins $\left( 2\times .25=50 \right)$ then there are $\dfrac{250-50}{50}=4$ half rupee coins.\[\]
 If there are 4 quarter rupee coins $\left( 4\times 0.25=100 \right)$ then there are $\dfrac{250-100}{50}=3$ half rupee coins. \[\]
If there are 6 quarter rupee coins $\left( 6\times .25=150 \right)$ then there are $\dfrac{250-150}{50}=2$ half rupee coins.\[\]
If there are 8 quarter rupee coins $\left( 8\times .25=200 \right)$ then there are $\dfrac{250-200}{50}=1$ half rupee coins. \[\]
So Latha got coins of 2 quarter rupee and 4 half rupees or 3 half rupees and 4 quarter rupee or 2 half rupee and 6 quarter rupee or 1 half rupee and 8 quarter rupee coins.

Note: If we take number of half rupee coins as $x$ and number quarter rupee coins as $y$ we can make linear equation $50x+25y=250\Rightarrow 2x+y=10$. The positive integral solutions of the equation will be the required result.