Question

Sonali has $67$ coins in his collection. There are three more $\mathrm{quarters}$than $\mathrm{nickels}$ and seven fewer $\mathrm{nickels}$ than $\mathrm{dimes}$. How many coins of each kind does Sonali have?

Answer 1

Step-1: Form equations using given data:

Given that,

• Sonali has $67$ coins in his collection
• Quarters are three more than nickels
• Nickels are seven fewer than dimes

Let the number of quarters be $\mathrm{x}$, number of nickels be $\mathrm{y}$ and the number of dimes be $\mathrm{z}$.

Then according to question, $\mathrm{x}+\mathrm{y}+\mathrm{z}=67$ $...\left(\mathrm{i}\right)$

And, $\mathrm{x}=\mathrm{y}+3$ $...\left(\mathrm{ii}\right)$

And $\mathrm{y}+7=\mathrm{z}$ $...\left(\mathrm{iii}\right)$

Step-2: Solve for the number of quarters, nickels and dimes:

On substituting $\left(\mathrm{ii}\right)$ and $\left(\mathrm{iii}\right)$ in $\left(\mathrm{i}\right)$, we get;

$\mathrm{y}+3+\mathrm{y}+\mathrm{y}+7=67$

$\Rightarrow$ $3\mathrm{y}+10=67$

$\Rightarrow$ $3\mathrm{y}=57$

$\Rightarrow$ $\mathrm{y}=19$

Substitute $\mathrm{y}=19$ in $\left(\mathrm{ii}\right)$ and $\left(\mathrm{iii}\right)$, we get;

$\mathrm{x}=19+3$

$\Rightarrow$$\mathrm{x}=22$

And, $19+7=\mathrm{z}$

$\Rightarrow$ $26=\mathrm{z}$

Hence, the number of quarters, nickels and dimes are $22,19$ and $26$, respectively.