Coin ratio after theft with Rs 2 and Rs 5 denominations

Key concept: Ratios of Rs 2 coins and Rs 5 coins become a solvable system once the remaining counts are compared after the theft.[1]

Step by step

Set up the coin counts

Let the original number of Rs 2 coins be $x$ and the number of Rs 5 coins be $y$. The problem says the total number of coins is 120, so we have

$x+y=120$

After the theft, one-third of the Rs 2 coins and one-quarter of the Rs 5 coins are removed. That leaves

$\frac{2}{3}x$ Rs 2 coins and $\frac{3}{4}y$ Rs 5 coins.

The new ratio is given as $8:15$, so

$\frac{\frac{2}{3}x}{\frac{3}{4}y}=\frac{8}{15}$

Solve the ratio equation

The expression $\frac{\frac{2}{3}x}{\frac{3}{4}y}$ simplifies to $\frac{8x}{9y}$. Setting that equal to $\frac{8}{15}$ gives

$\frac{8x}{9y}=\frac{8}{15}$

Cancel the common factor 8:

$\frac{x}{9y}=\frac{1}{15}$

Cross-multiplying gives

$15x=9y$

so

$5x=3y$.

Now use $x+y=120$. Since $y=\frac{5}{3}x$,

$x+\frac{5}{3}x=120$

$\frac{8}{3}x=120$

$x=45, \quad y=75$.

Find the original total value

The original value in the box is

$2x+5y=2(45)+5(75)=90+375=465$.

So the original monetary value was Rs 465.

Important consistency note

The extra statement about the stolen money being converted into Rs 10 notes does not match integer coin counts, because one-quarter of 75 is 18.75 coins, which is not possible for actual coins. The ratio information and total coin count still determine the original value as Rs 465, but the theft sentence is internally inconsistent as written.

Pitfall alert

The trap in this Rs 2 and Rs 5 coin problem is trying to use the stolen-money sentence first and getting stuck on fractional coins. The ratio after theft is the reliable equation, and it already determines the original counts. If you let the remaining Rs 2 coins be $\frac{2}{3}x$ and the remaining Rs 5 coins be $\frac{3}{4}y$, the ratio condition leads cleanly to $5x=3y$. Another common mistake is reading the 8:15 ratio backward as $15:8$, which would flip the system and produce the wrong coin counts. Also note that the given theft details are not fully consistent with whole coins, so do not force them to produce an exact Rs 10-note conversion if the arithmetic does not fit. The total original value comes from the valid system: 45 coins of Rs 2 and 75 coins of Rs 5, giving Rs 465.

Try different conditions

If the theft changed so that one-half of the Rs 2 coins and one-third of the Rs 5 coins were stolen, while the remaining ratio became 3:4, the same modeling method would apply with a new pair of equations. You would write $x+y=120$ and $\frac{\frac{1}{2}x}{\frac{2}{3}y}=\frac{3}{4}$. That kind of variation tests whether you can translate words into algebraic fractions without changing the structure of the system. If instead the coin types were Rs 1 and Rs 5, the ratio equation would still be linear, but the final value equation would change to $1x+5y$. In each variation, the key skill is converting the remaining fractions into a ratio equation before touching any value statement.

Further reading

ratio of remaining coins, system of linear equations, coin denomination value