# The value of a number of nickels and quarters is $3.25. If the number of nickels was increased by 3 and the number of quarters was doubled, the value would be$5.90. How do you find the number of each?

There are 10 quarters and 15 nickles needed to make $3.25 and$5.90 given the changes identified in the problem.
"The value of a number of nickels and quarters is $3.25" can then be written as: $0.05 n + 0.25 q = 3.25$This is because each nickle is worth 5 cents and each quarter is worth 25 cents. If the number of nickels was increased by 3 can be written as $n + 3$and "the number of quarters was doubled" can be written as $2 q$then the second equation can be written as: $\left(n + 3\right) 0.05 + 0.25 \left(2 q\right) = 5.90$or $0.05 n + 0.5 q = 5.75$Solving the first equation for $n$gives: $0.05 n + 0.25 q - 0.25 q = 3.25 - 0.25 q$$0.05 n = 3.25 - 0.25 q$$\frac{0.05 n}{0.05} = \frac{3.25}{0.05} - \frac{0.25 q}{0.05}$$n = 65 - 5 q$Substituting $65 - 5 q$for $n$in the second equation allows us to determine $q$or the number of quarters. $0.05 \left(65 - 5 q\right) + 0.5 q = 5.75$$3.25 - 0.25 q + 0.5 q = 5.75$$3.25 + 0.25 q - 3.25 = 5.75 - 3.25$$\frac{0.25 q}{0.25} = \frac{2.5}{0.25}$$q = 10$Substituting $10$for $q$in the first equation ($n = 65 - 5 q$) allows us to determine $n$or the number of nicklesL $n = 65 - 5 \cdot 10$$n = 65 - 50$$n = 15\$