Question

How many coins do you need to make $\$ 10$ if all the coins are quarters and if all the coins are dimes?

Answer 1

Hint:
Start with expressing the value of a Quarter coin and a Dime coin in cents. One dollar is equal to a hundred cents. Use the value in cents to find the number of Quarter and dime in one dollar. After obtaining these equations, multiply both sides with $10$ to get the required number of coins.

Complete step by step answer:
Here in this problem, we have coins in quarters and dimes. And we need to make $\$ 10$ of both quarters and dimes. The answer will be the number of coins of dimes and quarters to do so.
As we know that the dollar is a currency represented by a symbol $\$ $ and one dollar consists of hundred cents, i.e. $1\$ = 100{\text{ }}cents$.
The term “quarter” stands for the one-fourth part of a whole. Here, the quarter is referred to $\dfrac{1}{4}$ of a dollar or a hundred cents. Four of this coin carries the same value as one dollar
Therefore, $1{\text{ }}Quarter = \dfrac{1}{4} \times 100{\text{ }}cent = 25{\text{ }}cent$.
The term “dime” is used for one-tenth part of a dollar. This means ten dimes can be combined to represent one dollar.
Therefore, $1{\text{ dime}} = \dfrac{1}{{10}} \times 100{\text{ }}cent = 10{\text{ }}cent$
If we use coins of quarters to make ten dollars then:
As we know, we need four quarters to make $\$ 1$,
$ \Rightarrow 4 \times Quarter = 4 \times 25{\text{ }}cent = \$ 1$
And we need to make $10$ of those $\$ 1$,
$ \Rightarrow \left( {4 \times Quarter} \right) \times 10 = 10 \times \$ 1 = \$ 10$
Therefore, in solving this, we get:
$ \Rightarrow 40 \times Quarters = \$ 10$
Thus, we need $40$ Quarter coins to make $\$ 10$
If we use coins of dimes to make ten dollars then:
As we know, we need ten dimes to make $\$ 1$,
$ \Rightarrow 10 \times Dime = 10 \times 10{\text{ }}cent = \$ 1$
And we want to make $10$ of those $\$ 1$,
$ \Rightarrow \left( {10 \times Dime} \right) \times 10 = \$ 1 \times 10$
Now this can be solved further to get the required result
$ \Rightarrow 100 \times Dime = \$ 10$
Thus, we need $100$ Dimes to make $\$ 10$

Therefore, we’ll need $40$ Quarter coins and $100$ dime coins to make the required $\$ 10$.

Note:
In this question, the use of the values of Quarter and Dime in cents played a crucial role in the solution. An alternative approach can be to break the given amount of $\$ 10$ in form of Quarter value $\left( {25cents} \right)$ and dime value $\left( {10cents} \right)$. The value of $\$ 10$ can be expressed as: $\$ 10 = 10 \times \$ 1$ . Now you can use: $\$ 1 = 4 \times Quarter{\text{ and }}\$ 1 = 10 \times Dime$.