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.
If there are 4x 20¢ coins, there are 3x 50¢ coins.
20 20¢ coins = $4.00, or 8 50¢ coins.
So, we have
(4x-20)/(3x+8) = 7/11
x = 12
So, now you can figure the value of the money.
To solve this problem, we will follow these steps:
1. Set up equations to represent the given information.
2. Use the equations to solve for the number of coins.
3. Calculate the sum of money in the box.
Let's start with step 1:
Let's assume there are 4x twenty-cent coins and 3x fifty-cent coins in the box initially.
After 20 twenty-cent coins were taken out and replaced with fifty-cent coins:
The number of twenty-cent coins becomes (4x - 20).
The number of fifty-cent coins remains the same, which is 3x.
According to the given information, the ratio of the number of twenty-cent coins to the number of fifty-cent coins after the exchange is 7:11:
(4x - 20) / 3x = 7/11
Now, let's move on to step 2 and solve the equation:
11(4x - 20) = 7(3x)
44x - 220 = 21x
44x - 21x = 220
23x = 220
x = 220 / 23
x = 9.56522
Since we cannot have a fraction of a coin, we round x to the nearest whole number:
x = 10
Now, we can find the number of coins and the sum of money in the box:
Number of twenty-cent coins = 4x - 20 = 4(10) - 20 = 40 - 20 = 20
Number of fifty-cent coins = 3x = 3(10) = 30
The sum of money in the box can be calculated as follows:
Sum of money = (value of twenty-cent coins * number of twenty-cent coins) + (value of fifty-cent coins * number of fifty-cent coins)
Sum of money = (0.20 * 20) + (0.50 * 30)
Sum of money = 4 + 15
Sum of money = 19
Therefore, the sum of money in the box is $19.
To solve this question, we'll use algebra to set up equations based on the given information and then solve for the sum of money in the box.
Let's begin by assigning variables to the unknown quantities. Let's say the number of twenty-cent coins in the initial box is 4x (since the ratio is 4:3), and similarly, the number of fifty-cent coins is 3x.
After 20 twenty-cent coins are taken out and exchanged for fifty-cent coins, the number of twenty-cent coins decreases to 4x - 20, and the same number of fifty-cent coins (3x) is added back.
Now, we are given that the new ratio is 7:11, which means the new number of twenty-cent coins (4x - 20) should be in a ratio of 7 to the new number of fifty-cent coins (3x + 20), which should be in a ratio of 11.
Setting up the equation:
(4x - 20) / (3x + 20) = 7/11
We can now cross multiply to solve this equation:
11(4x - 20) = 7(3x + 20)
Simplifying the equation:
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 = 15
Now that we have the value of x, we can find the number of twenty-cent coins and fifty-cent coins in the initial box:
Number of twenty-cent coins = 4x = 4(15) = 60
Number of fifty-cent coins = 3x = 3(15) = 45
To find the sum of money in the box, we multiply the number of twenty-cent coins by 20 cents and the number of fifty-cent coins by 50 cents, and then add them together:
Sum of money = (Number of twenty-cent coins * 20 cents) + (Number of fifty-cent coins * 50 cents)
= (60 * 20) + (45 * 50)
= 1200 + 2250
= 3450 cents
Therefore, the sum of money in the box is 3450 cents.