259 nickels, 172 dimes and 69 quarters are present in the piggy bank.
According to the question, for every 2 nickels there dimes and for every 2 dimes there are 5 quarters.
[tex]2\;nickels\rightarrow 3\;dimes\\d\rightarrow \dfrac{2n}{3}[/tex]
and,
[tex]2\;dimes\rightarrow 5\;quarters\\q\rightarrow \dfrac{2d}{5}\\q\rightarrow \dfrac{4n}{15}[/tex]
And, a total number of coins in the piggy bank is 500.
So,
[tex]n+d+q=500\\n+\dfrac{2n}{3}+\dfrac{2}{5}\times \dfrac{2n}{3}=500\\\dfrac{5n}{3}+\dfrac{4n}{15}=500\\n=\dfrac{500\times 15}{25+4}\\n=\dfrac{7500}{29}\\n\approx 259[/tex]
Substitute [tex]n=\dfrac{7500}{29}[/tex] in [tex]d=\dfrac{2n}{3}[/tex],
[tex]d=\dfrac{2n}{3}\\d=\dfrac{2\times \frac{7500}{29}}{3}\\d=\dfrac{5000}{29}\\d\approx 172[/tex]
Again,
Substitute [tex]n=\dfrac{7500}{29}[/tex] in [tex]q=\dfrac{4n}{15}[/tex],
[tex]q=\dfrac{4n}{15}\\q=\dfrac{4\times \frac{7500}{29}}{15}\\q=\dfrac{2000}{29}\\q\approx 69[/tex],
Hence, 259 nickels, 172 dimes and 69 quartersare present in the piggy bank.
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