a child find 29 nickels and dimes. how many dimes are there if the total value of the coins are $2.10?
The correct answer and explanation is:
To find how many dimes there are, let’s use algebra.
Let:
- x be the number of nickels
- y be the number of dimes
We know two things:
- The total number of coins is 29:
x + y = 29 - The total value is $2.10, which is 210 cents. Nickels are worth 5 cents, and dimes are worth 10 cents:
5x + 10y = 210
Step 1: Solve the first equation for x
From x + y = 29, we get:
x = 29 − y
Step 2: Substitute into the second equation
Replace x in the value equation:
5(29 − y) + 10y = 210
145 − 5y + 10y = 210
145 + 5y = 210
Step 3: Solve for y
5y = 210 − 145
5y = 65
y = 13
Final Answer:
There are 13 dimes.
Explanation:
This is a classic coin word problem involving two variables: the number of coins and their total value. By assigning variables to each coin type and forming two equations based on the total quantity and total value, the problem becomes a system of equations. We solve this system using substitution.
The child found a total of 29 coins. Since nickels are worth 5 cents and dimes are worth 10 cents, we can express their total value in cents to avoid decimals. We rearranged the first equation to express one variable in terms of the other and substituted it into the second equation. This allowed us to solve for one variable, which represents the number of dimes. By solving, we found that the child has 13 dimes and the rest are nickels.