Using the Proportion Method to Solve Percent Problems

There are a variety of ways to solve percent problems, many of which can be VERY confusing. Fortunately, the PROPORTION METHOD will work for all three types of questions:

What number is 75% of 4?

3 is what percent of 4?

75% of what number is 3?


Using the PROPORTION METHOD, the set-up is always the same:

A proportion showing one fraction  with PART as the numerator and WHOLE as the denominator equal to another fraction with PERCENT as the numerator and 100 as the denominator.


PERCENT

-- the number with the percent sign (%).

PART

-- the number with the word is.

WHOLE

-- the number with the word of.

EXAMPLE #1:
What number is 75% of 4?   (or   Find 75% of 4.)
The PERCENT always goes over 100.
    (It's a part of the whole 100%.)
4 appears with the word of:
    It's the WHOLE and goes on the bottom.
A proportion showing one fraction  with PART as the numerator and 4 as the denominator equal to another fraction with 75 as the numerator and 100 as the denominator.
We're trying to find the missing PART (on the top).
In a proportion the cross-products are equal:   So 4 times 75 is equal to 100 times the PART.
The missing PART equals 4 times 75 divided by 100.
(Multiply the two opposite corners with numbers; then divide by the other number.)

4 times 75 = 100 times the part
300 = 100 times the part
300/100 = 100/100 times the part
3 =   the part
A proportion showing the denominator,  4, times the diagonally opposite 75; divided by  100.



EXAMPLE #2:
3 is what percent of 4?
3 appears with the word is:     It's the PART and goes on top.
4 appears with the word of:
    It's the WHOLE and goes on the bottom.
A proportion showing one fraction  with 3 as the numerator and 4 as the denominator equal to another fraction with percent as the numerator and 100 as the denominator.
We're trying to find the missing PERCENT (out of the whole 100%).
In a proportion the cross-products are equal:   So 3 times 100 is equal to 4 times the PERCENT.
The missing PERCENT equals 100 times 3 divided by 4.
(Multiply the two opposite corners with numbers; then divide by the other number.)

3 times 100 = 4 times the percent
300 = 4 times the percent
300/4 = 4/4 times the percent
75 =   the percent
A proportion showing the numerator,  3, times the diagonally opposite 100; divided by  4.



EXAMPLE #3:
75% of what number is 3?   (or 3 is 75% of what number?)
The PERCENT always goes over 100.
    (It's a part of the whole 100%.)
3 appears with the word is:
    It's the PART and goes on the top.
A proportion showing one fraction  with PART as the numerator and 4 as the denominator equal to another fraction with 75 as the numerator and 100 as the denominator.
We're trying to find the missing WHOLE (on the bottom).
In a proportion the cross-products are equal:   So 3 times 100 is equal to 75 times the WHOLE.
The missing WHOLE equals 3 times 100 divided by 75.
(Multiply the two opposite corners with numbers; then divide by the other number.)

3 times 100 = 75 times the whole
300 = 75 times the whole
300/75 = 75/75 times the whole
4 =   the whole
A proportion showing the numerator,  3, times the diagonally opposite 100; divided by  75.



Even unfriendly-looking problems like this can be solved using the PROPORTION METHOD:

EXAMPLE #4:
Find 83 2/3 % of 12.6   (or   What number is 83 2/3 % of 12.6?)
The PERCENT always goes over 100.
    (It's a part of the whole 100%.)
12.6 appears with the word of:
    It's the WHOLE and goes on the bottom.
A proportion showing one fraction  with PART as the numerator and 12.6 as the denominator equal to another fraction with 83 2/3 as the numerator and 100 as the denominator.
We're trying to find the missing PART (on the top).
In a proportion the cross-products are equal:   So 12.6 times 83 2/3 is equal to 100 times the PART.
The missing PART equals 12.6 times 83 2/3 divided by 100.
(Multiply the two opposite corners with numbers; then divide by the other number.)
12.6 times 83 2/3 = 100 times the part
(126/10)(251/3) = 100 times the part
31626/30 = 100 times the part
(31626/30)(1/100) = (100/1)(1/100)( the part)
31626/3000 =   the part
10542/1000 =   the part
10.542 =   the part
A proportion showing the denominator,  12.6, times the diagonally opposite 83 2/3; divided by  100.
PLEASE NOTE:   There are MANY other ways to do the arithmetic in this problem - hopefully this shows the steps in an understandable manner; it is neither the easiest nor the best approach.



The same method can be used to find the PART or the WHOLE if you are given a fraction instead of a percent. (Remember a percent is simply that amount per 100.).   Solve using cross-products:

SAMPLE PROBLEM #1:
  Find 3/4 of 20.  
 
Look at the set-up.

 
 
See the solution.





SAMPLE PROBLEM #2:
  10 is 2/3 of what number?
 
Look at the set-up.

 
 
See the solution.





  1. What is 20% of 60?

  2. 12 is 75% of what number?

  3. 6 is what percent of 8?

  4. 8 is 40% of what?

  5. 33 1/3% of what number is 24?

  6. Find 12.5% of 400.

  7. What is 5/8 of 32?

  8. 12 is 2/5 of what?

  9. What is 87 1/2% of 150?

  10. 25 is what percent of 30?
 

  1. 60 is 3/8 of what number?

  2. 12 is what fractional part of 20?

  3. 12 is what percent of 20?

  4. 9 is 37 1/2% of what?

  5. What is 0.25% of 30?

  6. 15 is what percent of 40?

  7. What percent of 60 is 75?

  8. 45 is 60% of what?

  9. Find 150% of 30

  10. Find 7/8 of 20.6

Additional practice exercises more percent practice problems



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