Question

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

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

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 $\mathrm{\$}10$ in form of Quarter value $\left(25cents\right)$ and dime value $\left(10cents\right)$. The value of $\mathrm{\$}10$ can be expressed as: $\mathrm{\$}10=10\times \mathrm{\$}1$ . Now you can use: $\mathrm{\$}1=4\times Quarter\text{ and }\mathrm{\$}1=10\times Dime$.