Question 149607
D-dimes
Q-quarters
H-half dollars
1.{{{D+Q+H=70}}} Total count of coins.
2.{{{0.10D+0.25Q+0.50H=17.75}}} Total value of coins.
3.{{{Q=3D}}} Three times dimes equals quarters.
Use eq. 3 to substitute for Q in eq. 1 and 2.
1.{{{D+Q+H=70}}}
{{{D+3D+H=70}}}
We'll call that equation 1a.
1a.{{{4D+H=70}}}
.
.
.
.
2.{{{0.10D+0.25Q+0.50H=17.75}}}
First multiply by 100 to get rid of decimals.
{{{10D+25Q+50H=1775}}}
{{{10D+25(3D)+50H=1775}}}
{{{10D+75D+50H=1775}}}
We'll call that equation 2a.
2a.{{{85D+50H=1775}}}
Use eq. 1a to come up with an expression for H as a function of D.
1a.{{{4D+H=70}}}
{{{H=70-4D}}}
Now substitute into eq. 2a,
2a.{{{85D+50H=1775}}}
{{{85D+50(70-4D)=1775}}}
{{{85D+3500-200D=1775}}}
{{{-115D=-1725}}}
{{{D=15}}}
From eq. 3,
{{{Q=3D}}}
{{{Q=3(15)}}}
{{{Q=45}}}
From 1a,
{{{H=70-4D}}}
{{{H=70-4*15}}}
{{{H=70-60}}}
{{{H=10}}}
There are 15 dimes, 45 quarters, and 10 half dollars.