SOLUTION: Jake has 205 coins consisting of nickels, dimes, and quarters. If the total value of the coins is $22 and there are three times as many dimes as nickels, how many of each coin did

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Question 224659: Jake has 205 coins consisting of nickels, dimes, and quarters. If the total value of the coins is $22 and there are three times as many dimes as nickels, how many of each coin did he have?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Jake has 205 coins consisting of nickels, dimes, and quarters. If the total value of the coins is $22 and there are three times as many dimes as nickels, how many of each coin did he have?

Step 1. Let x be the number of nickels.

Step 2. Let 0.05x be the dollar value of nickels.

Step 3. Let y be the number of dimes

Step 4. Let 0.10y be the dollar value of dimes.

Step 5. Let z be the number of quarters.

Step 6. Let 0.25z be the dollar value of quarters.

Step 7. Then x+y+z=205 since the total number of coins is 205.

Step 8. Then 0.05x+0.10y+0.25z=22 since the dollar value of nickels, dimes, and quarters is 22.

Step 9. We are also given y=3x since there are three times as many dimes as nickels.

Step 10. Here's our system of linear equations given in Steps 7, 8, and 9.

x%2By%2Bz=205 Equation A
0.05x%2B0.10y%2B0.25z=22 Equation B
y=3x Equation C

Substitute Equation C into Equations A and B

x%2B3x%2Bz=205 or 4x%2Bz=205 Equation A1

0.05x%2B0.10%283x%29%2B0.25z=22 or 0.35x%2B.25z=22 Equation B1

Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
We'll use substitution. After moving 1*z to the right, we get:
4%2Ax+=+205+-+1%2Az, or x+=+205%2F4+-+1%2Az%2F4. Substitute that
into another equation:
0.35%2A%28205%2F4+-+1%2Az%2F4%29+%2B+0.25%5Cz+=+22 and simplify: So, we know that z=25. Since x+=+205%2F4+-+1%2Az%2F4, x=45.

Answer: system%28+x=45%2C+z=25+%29.



With x=45 and z=25 then y=205-45-25=135

Check dollar value...45%2A0.05%2B135%2A0.10%2B25%2A0.25=+2.25%2B13.50%2B6.25=22...which is a true statement.

Step 11. ANSWER: The number of nickels is 45, the number of dimes is 135, and the number of quarters is 25.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J