Matthew was collecting small change for charity. He collected cents, nickels, and dimes. He collected one and a half times as many cents as nickels and 38 more dimes than nickels. There are 388 coins in his collection box. How much money did he collect altogether?
Let’s start with the three types of coins. Matthew collected cents, nickels, and dimes. We’ll let 𝑥 be the number of nickels that he collected. We’re told in the question that Matthew collected one and a half times as many cents as nickels. This is equal to 1.5𝑥 as one and a half is the same as 1.5. Matthew also collected 38 more dimes than nickels. As there were 𝑥 nickels, there’re 𝑥 plus 38 dimes.
Adding these three expressions gives us 1.5𝑥 plus 𝑥 plus 𝑥 plus 38. We were told that Matthew had 388 coins in his collection box. Therefore, this expression is equal to 388. Collecting the like terms on the left-hand side gives us 3.5𝑥 as 1.5𝑥 plus 𝑥 plus 𝑥 is 3.5𝑥.
We, therefore, have 3.5𝑥 plus 38 equals 388. Subtracting 38 from both sides of this equation gives us 3.5𝑥 is equal to 350 as 388 minus 38 equals 350. Dividing both sides of this equation by 3.5 gives us a value of 𝑥 equal to 100. As 𝑥 is equal to 100, we can now work out how many cents, nickels, and dimes Matthew had.
As the number of nickels was equal to 𝑥, Matthew has 100 nickels. 1.5 multiplied by 100 is equal to 150. Therefore, Matthew has 150 cents. 100 plus 38 is equal to 138. Therefore, Matthew has 138 dimes. We can check at this stage that these three numbers add up to 388. 150 plus 100 is 250. Adding 138 to this gives us 388.
We were asked to calculate how much money Matthew had altogether. In order to do this, we need to multiply 150 by 0.01. We need to multiply 100 by 0.05 and we need to multiply 138 by 0.1 or 0.10. This is because there are five cents in a nickel and there are 10 cents in a dime.
150 multiplied by 0.1 is equal to 1.5. Therefore, Matthew has one dollar and 50 cents worth of cents. 100 multiplied by 0.05 is equal to five. Therefore, Matthew has five dollars worth of nickels. 138 multiplied by 0.1 is equal to 13.8. Therefore, Matthew has 13 dollars and 80 cents of dimes.
In order to calculate the total amount of money that Matthew collected, we need to add one dollar and 50 cents to five dollars to 13 dollars and 80 cents. This is equal 20 dollars and 30 cents.
Altogether, Matthew has 20 dollars and 30 cents in his collection box. This is made up of cents, nickels, and dimes.