Emma has saved dollars, dimes and nickels in her money box. She has four more dollars than dimes and half as many nickels as she has dollars. She has saved 10 dollars and 85 cents in total. How many coins are there in her money box?
We need to work out how many dollars, how many dimes, and how many nickels Emma has. In order to do this, we’ll set up an equation using algebra. Our first step is to let the letter 𝑥 be the number of dollars that Emma has. As Emma has four more dollars than dimes, the number of dimes that she has is 𝑥 minus four as she has four less dimes than dollars. We’re also told that Emma has half as many nickels as she has dollars. Therefore, the number of nickels can be written as 𝑥 over two or 𝑥 divided by two.
At this stage, it is important to remember that a dime is worth 10 cents and a nickel is worth five cents. In order to work out the total value of these coins, we need to multiply the number of coins by the value. 𝑥 multiplied by one is equal to one 𝑥 or just 𝑥. 10 cents multiplied by 𝑥 minus four is equal to 0.1𝑥 minus 0.4.
We can show this by multiplying out the parenthesis below: 0.1 multiplied by 𝑥 minus four. 0.1 multiplied by 𝑥 is 0.1𝑥 and 0.1 multiplied by negative four is equal to negative 0.4. Finally, 𝑥 divided by two multiplied by five cents or 0.05 is equal to 0.025𝑥 as a half of 0.05 is 0.025.
We know from the question that the total value that Emma had saved was 10 dollars and 85 cents. We can, therefore, write the equation 𝑥 plus 0.1𝑥 minus 0.4 plus 0.025𝑥 is equal to 10.85. Simplifying the equation by grouping together our 𝑥 terms gives us 1.125𝑥 minus 0.4 equals 10.85. Adding 0.4 to both sides of this equation gives us 1.125𝑥 is equal to 11.25. Finally, dividing by 1.125 gives us a value for 𝑥 equal to 10.
The letter 𝑥 represented the number of dollars that Emma had. We can, therefore, say that Emma had 10 dollar bills. Emma had four more dollars than dimes. This means that she had six dimes as 10 minus four is equal to six. Finally, Emma had half as many nickels as she had dollars. This means that Emma had five nickels as a half of 10 is equal to five.
We can, therefore, calculate the total number of coins in her money box. We do this by adding 10, six, and five. This is equal to 21. Therefore, Emma had 21 coins in her money box.