Question

A collection of quarters and dimes is worth $\$12.40.$ There are 76 coins in all. How many of each coin are there?

Answer 1

Hint: This type of question is solved by converting the given word problem to a set of simultaneous equations. Then we can solve these equations to get the solution. We convert the dollars to cents and represent quarters and dimes in cents and form equations. We get one equation for the value of a quarter and a dime in cents. We get the second equation for the required number of dollars. Solving these simultaneously, we obtain the solution.

Complete step by step solution:
Let us the collection of quarters and dimes we have with us. Its total worth is $\$12.40.$ Dollar is a currency represented by the $\$$ symbol. Let us convert everything in terms of cents.
One dollar is equal to a hundred cents.
$\Rightarrow \$1=100\text{cents}$
Since we have $\$12.40$ dollars, this means that to convert this to cents, we need to multiply the value of 1 dollar with this amount we have.
$\Rightarrow \$12.40\times100=1240\text{cents}$
Let there be x quarters and y dimes that make up this 1240 cents. We can form an equation for this as,
$\Rightarrow x\times Quarter+y\times Dime=1240\text{ cents}\ldots \ldots \left( 1 \right)$
This forms our first equation. Now we know that one quarter is equal to one-fourth a dollar, or in terms of cents, it is equal to one-fourth of 100 cents.
$\Rightarrow 1Quarter=\dfrac{1}{4}\times 100\text{ cents}=25\text{ cents}$
Similarly, one dime is equal to one-tenth of a dollar or in terms of cents, this is equal to one-tenth of 100 cents.
$\Rightarrow 1Dime=\dfrac{1}{10}\times 100\text{ cents}=10\text{ cents}$
Substituting for the value of quarter and dime in equation (1) from above,
$\Rightarrow x\times 25+y\times 10=1240\ldots \ldots \left( 1 \right)$
Here everything is in terms of cents.
Also, it is given that there are a total of 76 coins. Therefore,
$\Rightarrow x+y=76\ldots \ldots \left( 2 \right)$
Now we have two equations, we need to solve them simultaneously to get the solution.
Multiplying both sides of equation (2) by 10,
$\Rightarrow 10x+10y=760\ldots \ldots \left( 3 \right)$
Subtracting the two equations (1) and (3),
$\begin{align}
  & \Rightarrow 25x+10y=1240 \\
 & \text{ }-10x-10y=-760 \\
 & \overline{\text{ +}15x+0y=480} \\
\end{align}$
We obtain an equation in terms of x only.
$\Rightarrow 15x=480$
Dividing both sides of the equation by 15,
$\Rightarrow \dfrac{15x}{15}=\dfrac{480}{15}$
We know that 32 times 15 is 480 and by cancelling out the terms,
$\Rightarrow x=32$
This is now substituted in equation (2),
$\Rightarrow 32+y=76$
Taking 32 to the other side,
$\Rightarrow y=76-32$
Subtracting 76 and 32,
$\Rightarrow y=44$
Since x represents the number of quarters and y represents the number of dimes, we have a total of 32 quarters and 44 dimes to make up $\$12.40.$
Hence, there are 32 quarters and 44 dimes in all.

Note: Currency conversion is an important part of this question and students are required to know the basics of this topic in order to solve this question easily. Another method to solve this is by trial and error, by taking different numbers of quarters and dimes and checking if it adds up to the total. But this method is tedious, hence its best to follow the approach given in this solution.