Samuel had some coins. The number of ten cent coins was 2/5 the number of twenty cent coins. Samuel took out 10 twenty cent coins from the bag and exchanged them for ten cent coins of equal value. The ratio of the number of ten cent coins to the number of twenty cent coins became 8/5. How many twenty cent coins and ten cent coins did he have at first?
Please help, thanks so much.
If there are x 10¢ coins and y 20¢ coins, then what do you know?
x = 2/5 y
after the switch, then
(x+20)/(y-10) = 8/5
(2/5 y + 20)/(y-10) = 8/5
2y+100 = 8y-80
now finish it off
Why did Samuel exchange his twenty cent coins for ten cent coins? Was he trying to make the Queen's head on the coins spin faster? 🤡
Alright, let's solve this brain teaser. Let's say Samuel originally had "x" number of twenty cent coins. According to the problem, he had 2/5 times the number of ten cent coins, which makes it 2/5x.
Now, he took out 10 twenty cent coins from the bag, which means he had x - 10 twenty cent coins left. He exchanged those ten twenty cent coins for ten cent coins, so he gained 10 ten cent coins.
After this exchange, the ratio of ten cent coins to twenty cent coins became 8/5. So we can set up the equation:
(2/5x + 10)/(x - 10) = 8/5
Now let's solve this equation and find the value of x. But, remember, solving this equation won't make you any richer, just a little bit smarter! 🤓
Let's assume that Samuel initially had 'x' number of twenty cent coins.
According to the information given, the number of ten cent coins is 2/5 of the number of twenty cent coins. Therefore, the number of ten cent coins is (2/5)x.
After taking out 10 twenty cent coins, Samuel has x-10 twenty cent coins remaining. He exchanged these 10 twenty cent coins for ten cent coins of equal value, which means he received 10 ten cent coins.
After this exchange, the ratio of the number of ten cent coins to the number of twenty cent coins became 8/5. We can set up the following equation:
(2/5)x + 10 = 8/5 * (x-10)
Let's solve this equation step by step:
Multiply both sides of the equation by 5 to get rid of the denomination in the denominators:
5 * (2/5)x + 5 * 10 = 5 * (8/5) * (x-10)
2x + 50 = 8 * (x-10)
Multiply 8 with the expression in parentheses:
2x + 50 = 8x - 80
Subtract 2x from both sides to isolate the variable:
2x - 2x + 50 = 8x - 2x - 80
50 = 6x - 80
Add 80 to both sides to move all variables to the right side:
50 + 80 = 6x - 80 + 80
130 = 6x
Divide both sides by 6 to solve for 'x':
130/6 = x
x ≈ 21.67
Since the number of coins cannot be in decimal, we'll round down to the nearest whole number:
x = 21
Therefore, Samuel initially had 21 twenty cent coins.
Now, we can find the number of ten cent coins:
Number of ten cent coins = (2/5)x = (2/5)(21) = 42/5 ≈ 8.4
We'll round down to the nearest whole number:
Number of ten cent coins = 8
Therefore, Samuel initially had 21 twenty cent coins and 8 ten cent coins.
To solve this problem, let's break it down step by step:
Step 1: Assign variables
Let's assign variables to the unknown quantities.
Let t be the number of ten cent coins Samuel had initially.
Let w be the number of twenty cent coins Samuel had initially.
Step 2: Set up the equations
Based on the given information, we can set up the following equations:
(1) The number of ten cent coins is 2/5 the number of twenty cent coins:
t = (2/5)w
(2) After exchanging 10 twenty cent coins, the ratio becomes 8/5:
(t + 10) / (w - 10) = 8/5
Step 3: Solve the equations
Now we can substitute the first equation into the second equation to solve for the values of t and w.
[(2/5)w + 10] / (w - 10) = 8/5
Simplifying this equation, we get:
(2w + 50) / 5(w - 10) = 8/5
Cross-multiplying, we get:
5(2w + 50) = 8(w - 10)
Simplifying further:
10w + 250 = 8w - 80
2w = -330
w = -165
Since the number of coins cannot be negative, we can conclude that there was an error in the problem setup or the data provided.
Please double-check the information and make sure it is accurate.