Algebra 3 methods of equations

Question

There are 35 coins, dimes and quarters. The coins are worth $6.65.

      How many coins of each kind are?

Solve the problem by using 3 methods( full explanations)

a)    Trial and error method

b)   Using equation method

c)    Using system of equations method

Answer 1

A. Trial and error - How would one get $6.65 with 35 coins only using dimes and quarters? If every 4 quarters is $1.00, you cannot exceed 28 quarters (or $7.00). So if you plugged in $5.00 worth of quarters, which is 20 quarters, how many dimes would be needed to make $6.65? Approximately 16 dimes, but you are left with 0.05 remaining. So this tells you that you would need an odd number of quarters (0.25 or 0.75) to get to a number ending in 5.

At 20 quarters ($5.00) and 16 dimes ($1.60) = $6.60, the total number of coins is 36.

Although the amount is not correct, the number of coins tells you that you need less coins.

If you need less coins, than the number of quarters should increase.

So the answer is...

At 21 quarters ($5.25) and 14 dimes ($1.40) = $6.65, and the number of coins is 35.

B. Equation - To make this problem into an equation.

d(0.10) + q(0.25) = 6.65

d = # of dimes

q = # of quarters

C. System of equations - To go further.

d + q = 35

If you solve for one variable, you could plug in for the other equation.

d + q = 35

d = 35 - q

If, d = 35 - q

Then, (35 - q)(0.10) + q (0.25) = 6.65

To simplify, 3.5 - 0.1q + 0.25q = 6.65

3.5 - 0.1q + 0.25q - 3.5 = 6.65 - 3.5

-0.1 q + 0.25 q = 3.15

0.15q = 3.15

0.15q/0.15 = 3.15/0.15

q = 21

Answer 2

Hi Stephen C.

14 dimes

21 quarters

There are several ways to approach the problem, since you are given the total number of coins and their total value. For trial and error you can evaluate pairs of numbers that sum to 35 when added together, you can use a linear equation and solve for y in terms of x in Slope Intercept Form (y = mx + b), you can use a System of Equations that can be solved by Elimination or Substitution and you can Graph the lines. I will go through one of the selections below you can evaluate the others

Let x = number of dimes

Let y = number of quarters

You are given the total number of coins, 35 so

x + y = 35

You are also given the total value, 6.65

0.10x + 0.25y = 6.65

Trial and error for (x,y) pairs of numbers that sum to 35

(20, 15), (21,14), (22, 13), to name a few to try in the total value equation

In Algebra there are 3 methods

Graphing a table of values such as the trial and error pairs above

Using an equation

Using a System of Equations (Elimination or Substitution)

Of course you can graph your equation and you can graph your system of equations too.

x + y = 35

y = -x + 35

For total value

0.10x + 0.25y = 6.65

y = -0.4x + 26.6

Since we have two equations for y we can set up an equation method

-x + 35 = -0.4x + 26.6

35 = x - 0.4x +26.6

35 - 26.6 = x - 0.4x

8.4 = 0.6x

8.4/0.6 = x

14 = x

y = -0.4 + 26.6

y = -0.4(14) + 26.6

y = -5.6 + 26.6

y = 21

For a system of equations there is more than one option

Substitution

.10x + .25(-x + 35) = 6.65

or

Elimination

10(.10x + .25y =6.65)

-(x + y = 35)

To give

x + 2.5y = 66.5

-x - y = -35

Of course there are other options. You can graph y = -x + 35 and y = -0.4x + 26.6. You should graph your equations to confirm the results.

Hope this helps.