Question

Lucy has \[34\] coins consisting of nickels and quarters amounting to \[2.90\]. How many coins of nickels and quarters does Lucy have?

Answer 1

Hint: In this problem we have given that Lucy has\[34\] coins but some are in nickels and some are in quarters. The value of nickels is $5$ cents ($5\% $ of a dollar) and the value of quarters is cents ($25\% $ of a dollar). The sum of the coins consisting of nickels and quarters is $34$.

Complete step by step answer:
Let us assume that the value of coins consisting of nickels island the value of coins consisting of quarters is y and we know that Lucy has $34$ coins.So,coins consisting of nickels coins consisting of quarters $ = 34$.
$x + y = 34............................................(1)$
The amount that Lucy has \[2.90\]. So, the next equation formed by considering the values of nickels and quarters is
$\dfrac{5}{{100}}x + \dfrac{{25}}{{100}}y = 2.90 \\
\Rightarrow 0.05x + 0.25y = 2.90 \\ $
After multiplying the whole equation with $100$, we get
$5x + 25y = 290$
Now, we will divide the whole equation by $5$, after dividing we get,
$x + 5y = 58 \\
\Rightarrow x = 58 - 5y............................(2) \\ $
Now, substitute the $(2)$ equation in $(1)$,
$58 - 5y + y = 34 \\
\Rightarrow 58 - 4y = 34 \\
\Rightarrow 58 - 34 - 4y = 0 \\
\Rightarrow 24 - 4y = 0 \\
\Rightarrow 24 = 4y \\
\Rightarrow\dfrac{{ 24}}{4} = y \\
\Rightarrow 6 = y \\ $
The value of coins consisting of quarters is $6$.Now substitute the value of y in equation$(1)$,
$x + y = 34 \\
\Rightarrow x + 6 = 34 \\
\Rightarrow x = 34 - 6 \\
\therefore x = 28 \\ $
Therefore, the value of coins consisting of nickels is $28$.

Note:The sum of the amount of coins consisting of $28$ nickels and the amount of coins consisting of $6$ quarters is \[2.90\] and the sum of the value of the coins is $ 34$. We can also check the amount of coins by adding,
The amount of $28$ nickels $ + $ the amount of $6$ quarters $ = 2.90$
$28 \times 0.05 + 6 \times 0.25 = 2.90 \\
\Rightarrow 1.4 + 1.5 = 2.90 \\ $
Hence, proved that the Lucy has $28$ coins of nickel and $6$ coins of quarters amounting to \[2.90\].