Stamp Word Problem

Question

Mason has a collection of 49cent stamps, 20 cent stamps, and 3 cent stamps worth $23.55. He had 56 total 49 cent and 20 cent stamps and the number of 3 cent stamps is nine more than the number of 20 cent stamps. How many of each does he have?

Answer 1

You need to write a bunch of equivalent expressions. Lets start with a few:

X= # of 49 cent stamps | Y = # of 20 cent stamps | Z = # of 3 cent stamps

All the stamps together make 23.55. I'm going to multiple all the numbers by 100 so they're easier to work with.

49X + 20Y + 3Z = 2355.

We also know the total number of 49 and 20 cent stamps is 56.

X + Y = 56

We also know the the number of 3 cent stamps is 9 more than the number of 20 cent stamps:

Z = Y + 9

Now that we have this, we want to make an expression with only one variable, so we are going to modify some expressions to be in terms of X.

The expression: [ X + Y = 56] is going to become [Y = 56 - X]

In the expression [Z = Y + 9] we are going to substitute the value of Y we just defined above to make

[Z = 56 - X + 9] and combine like terms to get [Z = 65 - X]

Now that we have defined both Y and Z in terms of X, we can plug these back into our general equation

[49X + 20Y + 3Z = 2355] will become [49X + 20(56- X) + 3(65 - X) = 2355]

Once you distribute the second equation, you can solve for X

[49X + 1120 - 20X + 195 - 3X] Combine like terms to get

[26X + 1315 = 2355]

[X = 40]

X =40, so there are 40, 49 cent stamps. We can plug this X value into our other equations to solve for Y and Z

[Z = 65 - X] becomes [Z = 65 - 40] Z = 25. There are 25 3 cent stamps

[Y = 56 -X] becomes [Y = 56 - 40] Y = 16. There are 16 20 cent stamps.

Hope this helps!

Answer 2

Hey Ariana, this is a gooood one!

Since Mason has 3 kinds of stamps, we have 3 variables.

I'm going to assign them:

49¢ stamps = x

20¢ stamps = y

3¢ stamps = z

Let's translate the word problem into math equations. (Hint: if you have 3 variables, then you'll need 3 equations to solve the problem. However many variables you have is how many equations you need.)

Translating the first sentence of the word problem:

Mason has a collection of 49 cent stamps, 20 cent stamps, and 3 cent stamps worth $23.55.

This one's a little tricky but we got this. Since we are dealing with worth or value, our equation is going to reflect that. We know how much xs, ys and zs are worth, so we multiply each variable by its worth, add 'em together and set it all equal to $23.55. Have a look:

.49x + .20y + .03z = 23.55 *Equation 1 of 3 complete!

Next one, translating the second part of the word problem:

He had 56 total 49 cent and 20 cent stamps...

Translate he had a total of 56 x stamps and y stamps altogether or...

x + y = 56 *Boom, equation 2 of 3 done.

Lastly, translating the third part of the word problem:

and the number of 3 cent stamps is nine more than the number of 20 cent stamps.

the number of 3 cent stamps means z

is means equals (=)

nine more than the number of 20 cent stamps means y + 9

or in other words:

z= y + 9

Alright, we have our 3 equations that we need in order to solve for our 3 variables:

Equation #1: .49x + .20y + .03z = 23.55

Equation #2: x + y = 56

Equation #3: z= y + 9

Now on to solving for one variable so we can solve for the other two so that we can find out how many of each stamp he has!

The gameplan is to use Equation #2 and Equation #3 in terms to y (since y is what they both have in common) and then plugging in those values of y into the equation #1. Check it out:

Equation #2: x + y = 56 or x = 56-y

and

Equation #3: z= y + 9

Let the plugging-in action into Equation #1 commence:

Equation #1:

.49x + .20y + .03z = 23.55

.49(56-y) + .20y + .03(y+9) = 23.55 *Now everything is in terms of y instead of 3 different variables.

27.44 -.49y +.20y +.03y +.27 = 23.55 * Distribute

-.26y +27.71 = 23.55 * Combine like terms.

-.26y = -4.16 * Isolate the variable

y = 16 or Mason has 16 of the 20¢ stamps

Now that you have y, use Equation #2 as x = 56-y and Equation #3 as z= y + 9 and solve for x and z!

x = 56-y

x = 56-16

x = 40 or Mason has 40 of the 49¢ stamps

z= y + 9

z= 16 + 9

z= 25 or Mason has 25 of the 3¢ stamps

So check it out,

49¢ stamps = x = 40

20¢ stamps = y = 16

3¢ stamps = z = 25

Does this help? Did you get the same answers? Can you rock a 3-variable, 3-equation word problem now? I hope so! Enjoy your stamps, Mason! I really hope that helps you out Ariana.

Cheers,

Natalie