Answer:
25 one-dollar coins, 16 half-dollar coins, and 164 quarters
Step-by-step explanation:
First, set up equations based on the information given:
[tex]0.25q+0.50h+1.00d=74[/tex]
[tex]\displaystyle{h=\frac{3}{5}d+1}[/tex]
[tex]q=4(d+h)[/tex]
Then, substitute q in the first equation with the expression from the third equation:
[tex]0.25[4(d+h)]+0.50h+1.00d=74\\1d+1h+0.50h+1.00d=74\\2d+1.5h=74[/tex]
Next, substitute h in that equation with the expression from the second equation:
[tex]\displaystyle{2d+1.5(\frac{3}{5}d+1)=74}[/tex]
[tex]2d+0.9d+1.5=74\\2.9d+1.5=74[/tex]
Solve for d, the number of one-dollar coins:
[tex]2.9d+1.5=74\\2.9d=72.5\\d=25[/tex]
Substitute 25 for d in the second equation to find h, the number of half-dollar coins:
[tex]\displaystyle{h=\frac{3}{5}d+1}[/tex]
[tex]\displaystyle{h=\frac{3}{5}(25)+1}[/tex]
[tex]h=15+1\\h=16[/tex]
Substitute 25 for d and 16 for h in the third equation to find q, the number of quarters:
[tex]q=4(d+h)\\q=4(25+16)\\q=4(41)\\q=164[/tex]
Then, verify that the coins total $74:
[tex]0.25(164)+0.50(16)+1.00(25)=74\\41+8+25=74\\74=74\\\text{Check.}[/tex]
Next, verify that the number of half-dollar coins is one more than three-fifths of the number of one-dollar coins:
[tex]\displaystyle{h=\frac{3}{5}d+1}[/tex]
[tex]\displaystyle{16=\frac{3}{5}(25)+1}[/tex]
[tex]16 = 15 + 1\\16 = 16\\\text{Check.}[/tex]
Finally, verify that the number of quarters is four times the number one-dollar and half-dollar coins together:
[tex]q=4(d+h)\\164=4(25+16)\\164=4(41)\\164=164\\\text{Check.}[/tex]
