Write a system of equations to represent the situation and solve. Define your variables.

Question

Malcolm has $9.60 in quarters and nickels for a total 100 coins. How many of each coin does he have?

Answer 1

q + n = 100

25q + 5n = 960

 

I think you can figure out where those equations came from.

 

First we need to solve the first equation for q or n. Let's solve for q by subtracting n from both sides.

 

q = 100 - n

 

Now, we take this "value" for q and substitute it into the 2nd equation.

 

25(100 - n) + 5n = 960

2500 - 25n + 5n = 960

-20n = -1540

n = 77

 

q = 100 - 77 = 23

 

Hope this helps!

Answer 2

Let q represent the number of quarters and n represent the number of nickels

 

The number of quarters + the number of nickels equals 100 coins, mathematically we have

 

q + n = 100

 

The value of a quarter is 0.25 dollars and the value of a nickel is 0.05 dollars so the amount of money in dollars for q quarters is 0.25q and the amount of money in dollars for n nickels is 0.05n.

 

The dollar amount for q quarters + the dollar amount for q quarters equal $9.60, mathematically we have

 

0.25q + 0.05n = 9.60

 

The system of equations is 

 

Eq 1:     q + n = 100

Eq 2:     0.25q + 0.05n = 9.60

 

We can solve this system using elimination, substitution or matrices.  I will use elimination.  The main idea for elimination is to choose a variable to eliminate and make the coefficients of that variable equal and opposite in signs by multiplying one or both equations by a nonzero number.  Then add the two equations.  Choosing q to eliminate we multiply the first equation by -0.25.  Remember to multiply each term of equation 1 by -0.025.  We have

 

-0.25q -0.25n =100(-0.25) giving -0.25q - 0.25n = -25 our system becomes

 

-0.25q - 0.25n = -25

0.25q + 0.05n = 9.60

 

Adding the two equations we have

 

-0.20n = -15.40

 

Divide each side by -0.20 gives

 

n = -15.40/-0.20 so n = 77

 

Now replace n by 77 in either equation 1 or 2 and solve for q.  I will use equation 1 because it is simpler than equation 2.  We have

 

q + 77 = 100

 

Subtract 77 from each side we have 

 

q = 100 - 77 so q = 23

 

He has 23 quarters and 77 nickels.

Answer 3

Let q = number of quarter and n = number of Nickels

 

First equation will represent the money and second equation will represent number coins

 

0.25q + 0.05n = 9.60

       q + n  = 100

 

0.25q +0.05n = 9.60 

-0.05(q + n = 100)   Multiply the whole equation by -.05 to eliminate n so that you can solve for q

 

0.25q + 0.05n = 9.60

-.05q  - 0.05n = -5.00  eliminate n and simply 

___________________

0.20q  = 4.60             re-write the decimal as fraction

(20/100)q = 460/100

20q = 460  (multiply both sides by 100 to cancel out the 100 in the denominator

q = 460/20  (divide both sides by 20)

q = 23

 

So he has 23 quarters. If there are 100 coins then he has 100 - 23 = 77 nickels 

 

Always and always check your solution. Substitute q and n in either equations

 

.25(23) + 0.05(77) = 9.60

 5.75 + 3.85 = 9.60

9.60 = 9.60 √

 

23 + 77 = 100 √