Coins...2

1. In a collection of dimes and quarters there are 6 more dimes than quarters. If there is $29.65 overall, how many of each are there?

Let q = quarters

Let d = dimes

I say q = 6 + d is one of the equations.
I saythe second equation is 0.10d + 0.25q = 29.65.

You say?

2. Zach had $11.85 in his piggy bank. He only had nickels and quarters in the bank. The number of quarters is equal to one less than twice the number of nickels. How many of each kind of coin did he have?

Let n = nickels.

Let q = quarters.

One of the equations is 0.05n + 0.25q = 11.85.

The second equation is q = 3(2n - 1).

You say?

 

I say that is wrong! If q= 6+ d then q, the number of quarters is 6 more than the number of dimes, the opposite of what you were told. The equation should be d= 6+ q;
Then the second equation become 0.10(6+ q)+ 0.25q= 0.60+ 0.10q+ 0.25q= 0.60+ 0.35q= 29.65 so 0.35q= 29.05. q= 29.05/0.35= 83. There are 83 quarters and 89 dimes.

Check: 83 quarters is worth 83(0.25)= $20.75. 89 dimes is worth 89(0.10)= $8,90. That is a total of $20.75+ $8.90= $29.65 as required.

If you didn't have such an aversion to actually completing the problem, you would have seen that "q= 6+ d" gives fractional values for d and q!

Again, NO! Where did that "3" come from? "Twice the number of nickels" is 2n and "one less than twice the number of nickels" is 2n- 1. q= 2n- 1.

Since q= 2n- 1, 0.05n+ 0.25q= 0.05n+ 0.25(2n- 1)= 0.05n+ 0.50n- 0.25= 0.55n- 0.25= 11.85. 0.55n= 11.85+ 0.25= 12.10. n= 12.10/0.55= 22. There are 22 nickels and 2(22)- 1= 44- 1= 43 quarters.

Check: "one less than twice the number of nickels", 22 nickels, is 43, the number of quarters. 22 nickels is worth 22(0.05)= $1.10. 43 quarters is worth 43(0.25)= $10.75. That gives a total of $1.10+ $10.75= $11.85, as required.

Again, had you completed the problem you would have seen that "q = 3(2n - 1)" would give fractional values for the number of quarters and nickels.

(Oh, and instead of "Let d= dimes", "Let q= quarters", and "Let n= nickels" it would be better to write, "Let d= the number of dimes", "Let q= the number of quarters", and "Let n= the number of nickels".)

 

What's up with the exclamation point?
Again, No! What? I guess you never make mistake.