How would you make 32 cents using dimes and pennies? And how would you make 32 cents using a quarter and other coins?
Both parts to this problem are all about making the amount 32 cents. But we’re limited in how we make it. In the first part of the problem, we’re told we can use dimes and pennies. We know that a dime has a value of 10 cents. And a penny, of course, is worth one cent. So really, what this question is asking is how can we make 32 using tens and ones. This is a place value question in disguise.
We can start by partitioning 32 into tens and ones. 32 is made up of three tens or 30 and also two ones. And we can make these two amounts using dimes and pennies. As we’ve said already, 30 is equal to three tens, which is the same as three 10 cent coins or three dimes. And the number two represents two ones. In coin terms, this is two pennies. We can make 30 cents out of dimes by having three dimes. And we can make two cents by having two pennies. And so, we can make 32 cents by adding three dimes and two pennies.
In the second part of the problem, we’re asked how we make 32 cents again, but this time using a quarter and other coins. We know that a quarter has a value of 25 cents. It’s a quarter dollar. We can use this fact to help us draw another part whole model. These models are useful in helping us to work out the answer. So we start off with the number 32. And we’re going to partition it in a different way this time. We’re told that one of the coins we need to use is a quarter. So one of the numbers that we partition 32 into is 25.
Let’s write the coin that we’re using next to it. 25 is the same as one quarter. Now, the difference between 32 and 25 is seven. And so, our remaining part is seven. We need to make up seven cents somehow. How can we do this? Well, we could use seven pennies. But there’s another way to make seven cents using less coins. We know a nickel has a value of five cents. We could partition the number seven into five and two. As we’ve just said, we could use one coin to make five cents and that’s a nickel. And then to make our remaining two cents, we’d need two pennies. So to make 32 cents, we’ve used one quarter, one nickel, and two pennies.
In both parts of this problem, we’ve made 32 cents. We’ve made it in different ways. To make 32 cents using dimes and pennies, we thought about the number of tens and ones that there are in 32. We could make 32 cents then by adding three dimes and two pennies. In the second part, the problem was opened up to include a quarter and other coins. We know that a quarter is worth 25 cents. And 25 plus seven equals 32. So by using a nickel and two pennies, we could make the remaining seven cents.
So by using a quarter and other coins, we could make 32 cents by adding one quarter, one nickel, and two pennies. We can make the same amount of money in different ways, using different coins.