A coin box contained some twenty-cent and fifty-cent coins in the ratio 4 : 3. After 20 twenty-cent coins were taken out to exchange for fifty-cents coins of the same value and put back in the box, the ratio of the number of twenty-cent coins to the number of fifty-cents coins became 7 : 11. Find the sum of money in the box.
I posted this question before and based on formula below my final calculation did not match the given answer which is $27.60
(4x-20)/(3x+8) = 7/11
x = 12
Please help!
number of 20 cent coins ---- 4x
number of 50 cent coins ---- 3x
(notice 4x : 3x = 4 : 3)
20 of the twenty cents coins were taken out
so we have 4x - 20 left
value of those 20 coins = 20(20) = 400 cents
which would make 400/50 or 8 fifty cent pieces
So number of 50 cent coins = 3x + 8
(4x-20)/(3x+8) = 7/11
44x - 220 = 21x + 56
23x = 276
x = 12
So originally there were 4(12) or 48 twenty cent pieces
and 2(12) or 36 fifty cent pieces.
total amount = 48(20cents) + 36(50cents) = 2760 cents or $27.60
To solve this problem, we can start by using the information given to set up an equation.
Let's assume that the original number of twenty-cent coins is 4x, and the original number of fifty-cent coins is 3x.
After 20 twenty-cent coins are taken out and exchanged for fifty-cent coins, the new number of twenty-cent coins becomes 4x - 20, and the new number of fifty-cent coins becomes 3x + 20. We add 20 because 20 twenty-cent coins were exchanged.
Given that the ratio of the number of twenty-cent coins to the number of fifty-cent coins after the exchange is 7:11, we can set up the equation:
(4x - 20) / (3x + 20) = 7/11.
Now, we can solve this equation for x.
Cross-multiplying the equation, we get:
11(4x - 20) = 7(3x + 20).
Expanding both sides of the equation, we get:
44x - 220 = 21x + 140.
Moving like terms to one side of the equation, we have:
44x - 21x = 140 + 220.
Combining like terms, we get:
23x = 360.
Dividing both sides of the equation by 23, we find:
x = 360 / 23.
Now, we need to find the sum of money in the box. To calculate this, we first find the value of each coin.
The value of a twenty-cent coin is $0.20, and the value of a fifty-cent coin is $0.50.
We know that the original number of twenty-cent coins is 4x, which is 4 * (360/23) = 240/23.
So, the value of the original twenty-cent coins is (240/23) * $0.20.
Similarly, the original number of fifty-cent coins is 3x, which is 3 * (360/23) = 540/23.
So, the value of the original fifty-cent coins is (540/23) * $0.50.
Now, we can calculate the sum of money in the box:
Sum of money = value of original twenty-cent coins + value of original fifty-cent coins.
Sum of money = (240/23) * $0.20 + (540/23) * $0.50.
Simplifying this expression, we get:
Sum of money = $8.70 + $11.79.
Sum of money = $20.49.
Thus, the sum of money in the box is $20.49.
To solve this problem, let's break it down step by step:
1. Let's represent the number of twenty-cent coins and fifty-cent coins in the box as 4x and 3x respectively. The ratio is given as 4:3.
2. After 20 twenty-cent coins are taken out, we have (4x - 20) twenty-cent coins left.
3. Now, we exchange these 20 twenty-cent coins for fifty-cent coins. Each twenty-cent coin is exchanged for one fifty-cent coin of the same value. So, we add 20 fifty-cent coins to the box, resulting in (3x + 20) fifty-cent coins.
4. After the exchange, the ratio of twenty-cent coins to fifty-cent coins becomes 7:11, which can be written as (4x - 20)/(3x + 20) = 7/11.
Now, let's solve this equation to find the value of x:
(4x - 20)/(3x + 20) = 7/11
Cross-multiplying, we get:
11(4x - 20) = 7(3x + 20)
44x - 220 = 21x + 140
Subtracting 21x from both sides:
23x - 220 = 140
Adding 220 to both sides:
23x = 360
Dividing both sides by 23:
x = 360/23
Now, to find the sum of money in the box, we need to calculate the total value of the coins.
Total value = (value of twenty-cent coins) + (value of fifty-cent coins)
Value of twenty-cent coins = (number of twenty-cent coins) * 20 cents = (4x - 20) * 20 cents
Value of fifty-cent coins = (number of fifty-cent coins) * 50 cents = (3x + 20) * 50 cents
Substituting the value of x we found earlier:
Value of twenty-cent coins = (4 * (360/23) - 20) * 20 cents
Value of fifty-cent coins = (3 * (360/23) + 20) * 50 cents
Now, calculate the total sum of money in the box by adding the two values together:
Total sum of money = (Value of twenty-cent coins) + (Value of fifty-cent coins)
Finally, convert the total sum to dollars by dividing it by 100 cents:
Total sum in dollars = (Total sum of money) / 100
By following these steps and calculations, you should be able to find the correct answer.