A shopkeeper sold a certain number of toys.The number of toys as well as da price of each toy in Rs. was a two digit number. By mistake he reversed da digits of both the number of toys he sold and the price of each toy.As a result, he found that his stock account at the end of the day showed 81 items more than actual present price
original number ---- 10x + y, where x is the tens digit and y is the unit digit
10x + y = 18 or y = 18-10x
reversed:
10y + x = 81
solving by substitution, ....10(18-10x) + x = 81
180 - 100x + x = 81
-99x = -99
x = 1
then y = 18-10 = 8
He sold 18 toys at RS 18 each
so each toy cost RS 1
then 81 would cost 81
correct
Let's break down the information given:
1. The number of toys sold and the price of each toy are both two-digit numbers.
2. The shopkeeper made a mistake by reversing the digits of both the number of toys sold and the price of each toy.
3. The shopkeeper's stock account at the end of the day showed 81 items more than the actual present price.
Let's assign variables to the number of toys sold and the price of each toy:
Number of toys sold = 10a + b (a is the tens digit, b is the ones digit)
Price of each toy = 10c + d (c is the tens digit, d is the ones digit)
According to the given information, the reversed numbers are:
Reversed number of toys sold = 10b + a
Reversed price of each toy = 10d + c
The equation based on the shopkeeper's stock account is:
Reversed number of toys sold - Reversed price of each toy = Actual present price + 81
Substituting the values of the reversed numbers:
(10b + a) - (10d + c) = 10a + b + 81
Expanding and simplifying the equation:
10b + a - 10d - c = 10a + b + 81
9b - 9d - c = 9a + 81
To solve this equation, we can try different values for a, b, c, and d until we find a solution where all values are between 0 and 9 and the equation holds true.
To solve this problem, let's break it down into smaller steps:
Step 1: Understand the problem.
We need to find the number of toys sold and the price of each toy, given that the digits of both numbers were reversed. We also know that the shopkeeper's stock account at the end of the day showed 81 items more than the actual present price.
Step 2: Identify the unknowns.
Let's assume the original number of toys sold was X, and the original price of each toy was Y. We need to find the values of X and Y.
Step 3: Formulate equations.
Based on the given information, we can establish two equations:
1) When the digits are reversed, the new number of toys sold is 10Y + X, and the new price of each toy is 10X + Y.
2) The stock account at the end of the day shows 81 items more than the actual present price, so the equation becomes: 10Y + X = 10X + Y + 81.
Step 4: Solve the equations.
Now we can solve the equations simultaneously to find the values of X and Y.
10Y + X = 10X + Y + 81
9Y - 9X = 81
Y - X = 9 (Dividing both sides by 9)
From the equation Y - X = 9, we know that the difference between Y and X is 9. Since both X and Y are two-digit numbers, there are a limited number of possible values. We can list them:
- X = 10, Y = 19
- X = 11, Y = 20
- X = 12, Y = 21
- X = 13, Y = 22
- X = 14, Y = 23
- X = 15, Y = 24
- X = 16, Y = 25
- X = 17, Y = 26
- X = 18, Y = 27
- X = 19, Y = 28
- X = 20, Y = 29
- X = 21, Y = 30
We can now go through each possibility and apply the second equation to find the correct values.
Step 5: Find the correct values.
Based on the second equation, 10Y + X = 10X + Y + 81, we can substitute the values from each possibility to check if it satisfies the equation. By doing this, we find that X = 18 and Y = 27 is the solution.
Therefore, the original number of toys sold was 18, and the original price of each toy was Rs. 27.