SOLUTION: The difference of two numbers is 16. Three-fourths of the larger number is 14 larger than one-half of the smaller number. Find the numbers.

Question 204036: The difference of two numbers is 16. Three-fourths of the larger number is 14 larger than one-half of the smaller number. Find the numbers.
Found 2 solutions by ankor@dixie-net.com, PRMath:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Two numbers, x & y
;
The difference of two numbers is 16.
x - y = 16
x = (y+16)
:
Three-fourths of the larger number is 14 larger than one-half of the smaller number.
.75x = .5y + 14
:
Replace x with (y+16)
.75(y + 16) = .5y + 14
:
.75y + 12 = .5y + 14
:
.75y - .5y = 14 - 12
:
.25y = 2
y =
y = 8
then
x = 8 + 16
x = 24

Answer by PRMath(133) (Show Source): You can put this solution on YOUR website!
There are probably a lot of equations you could use but I used this:

First statement:

The difference of two numbers is 16. Ok, let's call the first number the LARGER number and the 2nd number the SMALLER. We'll label those as "L" (for Larger) and "S" (for Smaller).

SO... again:

The difference of two numbers is 16.
Translation: L - S = 16

Second Statement:
Three-fourths of the larger number.
Translation: L

Third statement:
is.
Translation: =

Fourth statement:
14 larger than one-half of the smaller number.
Translation: 14 + S

Now put all our translations together to see our system:

L - S = 16
L = 14 + S

Now let's multiply the entire 2nd equation by 4 to get rid of the denominators.

If we multiply the above by 4, we'll get:

3L = 56 + 2S (now let's divide everything by "3" to isolate the "L"
= + (dividing everything by 3)
L = + S

Now let's go back to our original equation of: L - S = 16. We can substitute in the value of "L" into this equation. Try this:

L - S = 16
+ S - S = 16 (see how we put this + S in for L?)
56 + 2S - 3S = 48 (I multiplied the whole line times 3 to get rid of the fraction)
56 - S = 48 (combined the "S" terms)
-S = 48 - 56 (subtracted both sides by 56 to isolate the S)
-S = -8 (subtracted 48 - 56)
S = 8 (divided both sides by -1 to further isolate the S)

SO the smaller number, which is "S" = 8. Let's go back to our original equation again.....

L - S = 16 (original equation)
L - 8 = 16 (inserted the value of "S" (which is 8) for the variable "S")
L = 16 + 8 (added 8 to both sides to isolate the "L)
L = 24 (added 16 + 8).

SO now we know the smaller number is 8 and the larger number is 24. Does it check out with the problem?

Well, 24 - 8 = 16, so our first part of the problem is right.

What about the 2nd part of the problem?

Three-fourths of the larger number: (24)
is: =
14 larger than one-half of the smaller number: 14 + S

SO (24) = 14 + 8 Let's do the math....
= 14 + 4

Seems we have it checked out.

I hope this helps. :-)
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