Katie has a collection of nickels, dimes and quarters with a total value of $4.90. There are 7 more dimes than nickels and 4 more quarters than nickels, how man

Question

Katie has a collection of nickels, dimes and quarters with a total value of $4.90. There are 7 more dimes than nickels and 4 more quarters than nickels, how many of each coin does she have?

Answer 1

Hi Hanna-- So you know that some number of nickels, dimes and quarters will add up to 490 cents. Every nickel is worth 5 cents, so some number of nickels can be 5n. In the same way, some number of dimes can be 10d, and some number of quarters can be 25q. So the general equation is going to be 5n + 10d + 25q = 490 But since the dimes and quarters are also explained in relation to the number of nickels, think of everything as nickels: the number of dimes (d) = the number of nickels + 7, so d = n + 7 And in the same way, you can arrive at the number of quarters (q) is nickels + 4, so q = n + 4 So re-write the equation, but each time you see d, write n + 7 instead and each time you see q, write n + 4. you'll end up with lots of chunks of n: 5n + 10(n + 7) + 25(n + 4) = 490 when you do all the multiplying, you get: 5n + 10n + 70 + 25n + 100 = 490 combine all those terms and you get 40n + 170 = 490. Subtract 170 from both sides and you get 40n = 320. Divide both sides by 40 and you get n = 8. Which means you have 8 nickels, then add 7 to get 15 dimes, and for the quarters it was the nickels plus 4, so you have 12 quarters. When you double check by multiplying out how much each pile of coins is worth, it will add up to 490 cents, or $4.90 Fun, right? :-) Matt

Answer 2

Katie,

 

Let the number of nickles = N

 

Since there are 7 more dimes than nickles,

 

the number of dimes D = N + 7

 

Since there are 4 more quarters than nickles,

 

the number of quarters Q = N + 4

 

Since quarters = $0.25, dimes = $0.10, and nickles = $0.05 and all added together = $4.90 ⇒ 

 

0.25·Q + 0.10·D + 0.05·N = 4.90

 

Now multiply through by 100 to eliminate the decimal:

 

25Q + 10D + 5N = 490

 

Substitute for Q and D

 

25(N+4) + 10(N+7) + 5N = 490

 

25N + 10N + 5N +100 +70 = 490

 

40N + 170 = 490 ⇒ 40N = 320 ⇒ N = 320/40

 

N = 8,

 

D = N+7 = 15

 

Q = N+4 = 12

 

Check:

 

25(12) + 10(15) + 5(8) = 300 + 150 + 40 = 490

 

 

 

 

 

 

 

 

Answer 3

Hanna:

 

Let x be Katie's nickels. Then her dimes are x + 7 and her quarters are x +4.  Her total cash i(in terms of cents) s

 

5x + 10(x + 7) + 25 (x + 4) = 40x + 170.

 

This total is given to be $4.90 = 490 cents.  Equate the two.

 

40x + 170 = 490

 

40x = 320

 

x = 320/40 = 8

 

Katie has x = 8 nickels (40c), x + 7 = 15 dimes (150c), and x + 4 = 12 quarters (300c), adding up to 490c = $4.90

 

Dr. G.