Question

A collection of dimes and nickels is worth $5.60. The ratio of dimes to nickels is 3:2. How many dimes and how many nickels are there?

Answer 1

Let

x------> the number of dimes

y-----> the number of nickels

we know that

$1\ dime=\ 0.10$

$1\ nickel=\ 0.05$

so

$0.10x+0.05y=5.60$ ------> equation A

$\frac{x}{y} =\frac{3}{2}$

$x=1.5y$ -----> equation B

substitute equation B in equation A

$0.10[1.5y]+0.05y=5.60$

$0.20y=5.60$

$y=28\ nickels$

Find the value of x

$x=1.5y$

$x=1.5*28=42\ dimes$

therefore

the answer is

the number of dimes is $42$

the number of nickels is $28$

Answer 2

Now, the nickels and dimes are at ratio of 3:2.  However, the total amount is in dollars.  So let's convert all amounts to pennies then.

$5.60 is just 560 pennies

3 dimes is just 30 pennies

2 nickels is just 10 pennies

so, the ratio from dimes to nickels is a ratio of 30:10 pennies, or 3:1 then, for a total amount of 560 pennies.

$\bf \cfrac{dimes}{nickels}\qquad 3:2\qquad\cfrac{\stackrel{pennies}{30}}{\stackrel{pennies}{10}}\implies \cfrac{30}{10}\implies \cfrac{3}{1}\implies 3:1~ratio \\\\\\ \cfrac{\stackrel{dimes}{pennies}}{\stackrel{nickels}{pennies}}\qquad 3:1\qquad \cfrac{3}{1}\implies \cfrac{3\cdot \frac{560}{3+1}}{1\cdot \frac{560}{3+1}}\implies \cfrac{3\cdot 140}{1\cdot 140}\implies \cfrac{420}{140}\\\\ -------------------------------$

$\bf \textit{now, how many dimes in 420 pennies?}\quad \cfrac{420}{10}\implies 42~dimes \\\\\\ \textit{and how many nickels in 140 pennies?}\quad \cfrac{140}{5}\implies 28~nickels\\\\ -------------------------------\\\\ 42~dimes + 28~nickels\implies \ 5.60$