FIGURING PERCENTAGES OF A NUMBER
1. Divide the percentage by 100 to convert it to a decimal.
2. Multiply the decimal by the number.
3. Ask yourself: Is this answer logical, or could I have made a math error?
Example: What is 23 percent of 400?
1. 23 divided by 100 = .23
2. .23 x 400 = 92
3. Is this answer logical? Yes! One-fourth (25 percent) of 400 would be 100, so 23 percent of 400 should be just less than 100. Our answer, 92, is just less than 100.
FIGURING THE PERCENTAGE WHEN YOU KNOW TWO NUMBERS
1. Divide one number by the other.
2. Multiply the result by 100.
3. Ask yourself: Is this answer logical, or could I have made a math error?
Example: Thirty-eight is what percentage of 400?
1. 38 divided by 400 = .095
2. .095 x 100 = 9.5 percent. Thirty-eight is 9.5 percent of 400.
3. Is this answer logical? Yes! Ten percent of 400 would be 40 (you just lop off a zero), so our answer for 38 should be nearly 10 percent. It is.
FIGURING A WEIGHTED AVERAGE
1. Divide the percentages by 100 to get decimals.
2. Multiply the decimals by the appropriate scores.
3. Add the products to get the final score.
Example: In History of American Civilization, there are three tests, each of which counts 20 percent of a student’s final grade, and a final paper, which counts 40 percent. Mary’s test grades were 83, 50 and 76, and she made an 80 on the paper. Can Mary earn a B for the course? (Her average would have to be at least 79.5.)
1. Divide the percentages by 100 to get decimals. Twenty percent becomes .20, and 40 percent becomes .40.
2. Multiply the scores by the appropriate decimal expression, like this:
83 x .20 = 16.6
50 x .20 = 10
76 x .20 = 15.2
80 x .40 = 32
3. Add the numbers to get the final score: 16.6 + 10 + 15.2 + 32 = 73.8. Mary will make a C.
FIGURING PERCENTAGE INCREASE
1. Subtract the old number from the new number to get the amount of change.
2. Divide the amount of change by the old number.
3. Multiply by 100
4. As yourself: Is this answer logical, or could I have made a math error?
Example: On the first leg of the trip, John and Susan faced repeated traffic jams and averaged a speed of only 30 mph. On the second leg of the trip, traffic was lighter and they averaged 50 mph. By what percentage did their speed increase?
1. 50 – 30 = 20
2. 20 divided 30 = .667
3. .667 x 100 = 66.7 percent. John and Susan’s average speed was 66.7 percent higher on the second leg of the trip.
4. Is this answer logical? Yes, 50 mph is significantly more than 30 mph.
Note: Problems that don’t involve actual increases can be worked using this formula. For example, you could use this method to solve this problem: “Elise works six hours a week; her sister, Elaine, works 20. How much more does Elaine work than Elise?” Here's how to solve that problem: 1) 20 - 6 = 14; 2) 14 divided by 6 = 2.333; 3) 2.333 times 100 = 233.3. Elaine works 233.3 percent more hours than Elise. Is this logical? Yes, because Elaine works more than twice as much (200 percent) than Elise.
FIGURING PERCENTAGE DECREASE
1. Subtract the new number from the old number to get the amount of change.
2. Divide the amount of change by the old number
3. Multiply by 100
4. Ask yourself: Is this answer logical, or could I have made a math error?
Example: Tinytown’s tax revenues are declining, so it reduced the amount of money it spends on building maintenance from $40,000 a year to $32,800 a year. By what percentage did Tinytown’s building maintenance expenditures fall?
1. $40,000 - $32,800 = $7,200
2. $7,200 divided by $40,000 = .18
3. .18 x 100 = 18 percent, the amount by which Tinytown’s building maintenance expenditures fell
4. Is this answer logical? Yes! Twenty-five percent of $40,000 — an easy number to figure — would have been $10,000, and $7,200 is a little less than $10,000. So our answer should be a little less than 25 percent. Eighteen percent is a little less than 25 percent.
FIGURING NEW NUMBER OR AMOUNT OF CHANGE FROM PERCENTAGE AND OLD NUMBER
1. Divide the percentage by 100 to turn it into a decimal
2. Multiply the old number by the decimal. This gives you the amount of change.
3. Subtract the amount of change from the old number. This gives you the new number.
Example: John Smith moved from Casa Grande to Gilbert and cut his commute to All State University, which had been 50 miles one-way, by 70 percent. How far does he commute now?
1. 70 divided by 100 = .70
2. 50 x .70 = 35, the difference between the old and new distances
3. 50 – 35 = 15, the new distance that John commutes now
4. Ask yourself: Is this logical? Yes! John reduced his commute by nearly three-quarters, and 35 is nearly three-quarters of 50.
FIGURING OLD NUMBER OR AMOUNT OF CHANGE FROM PERCENTAGE INCREASE AND NEW NUMBER (Hold on; this is a little tricky!)
1. Divide the percentage by 100 to turn it into a decimal
2. Set up an equation in which Y stands for the old number. The new number, which you know, is the old number (Y) plus some percentage of Y (the percentage increase). So you can say:
Y + (Y times percentage increase expressed as a decimal) = new number
3. Because Y = 1Y, you can change the left half of the equation to a single number (1 plus the percentage increase expressed as a decimal) times Y. If your percentage increase were 30 percent, the equation would look like this:
1.30Y = new number
4. To get the Y on a side by itself, divide both sides by the number before Y, in this case 1.30.
Example: Tanya Lopez makes $1,000 a week in her new job, 10 percent more than she made in her old job. What was her old salary?
1. 10 divided by 100 = .10
2. Let Y stand for Tanya’s old salary. You know that Y + .10Y = $1,000
3. 1.10Y = $1,000
4. Y = $1,000 divided by 1.10
5. Y = $909.09
6. Ask yourself: Is this logical? Yes! Tanya’s salary increased by 10 percent, and 10 percent of 909.09 would be $90.90. Add that to $90.90 and you get $999.99, within a penny of $1,000.
FIGURING OLD NUMBER OR AMOUNT OF CHANGE FROM PERCENTAGE DECREASE AND NEW NUMBER. (This is the toughest challenge!)
Example: John Smith moved from Casa Grande to Gilbert and cut his commute to All State University by 70 percent, to 15 miles. How far did he commute before he moved? Let’s pretend we don’t already know the answer from the problem above and reach back into the recesses of our memories about ninth-grade algebra. We know that when John reduced his commute, he cut it by 70 percent. So we can deduce that his remaining commute is 30 percent of his old journey (100 – 70 = 30). Just for convenience sake, let’s call his old journey length Y. We know that 70 percent of his old journey length added to his new journey length would equal his old journey. If we wanted to express that algebraically, we could do it like this: 70 percent of the old journey + the new journey (30 percent of the old journey) = 100 percent of the old journey
OR
1. (Y x .70) + 15 = Y
2. .70Y + 15 = Y Get all the Y’s on one side by subtracting .70Y from each side.
3. 15 = Y - .70Y
4. 15 = .30Y Get rid of .30 by dividing both sides by .30.
5. 15 divided by .30 = 50
Below are some scenarios that reflect the kind of question typically faced by journalists and frequently answered incorrectly. They were created by John Oudens, a professional copy editor who is married to Dr. Keith. The answers are at the end.
1. In 2001 the city’s emergency fund stood at $900,000, 50 percent larger than it was the year before, and the money largely went unspent while many city departments had to cut costs. This year, the city council has decided that the emergency fund need only be 25 percent larger than it was in 2000. How much money does the council want the fund to have now, in dollars?
2. “Moose” gave up team sports after he graduated from high school, but his appetite stayed the same, and he’s up to 300 pounds. He is determined to drop his weight to 240. As a percentage, how much of his weight does he intend to lose?
3. Smallville College’s total enrollment was 460 students two years ago. Now the enrollment is 483. What is the percentage increase from two years ago?
4. I can drive 575 miles in a day. That is equal to 20 percent of a trip I have to make next week. How long is the trip and how many days will it take?
5. In a metropolitan area comprising 2 million homes, 1.5 million homes have cable television, and an additional 300,000 homes have satellite dishes. The other homes have neither. As a percentage, how many of the metropolitan’s area homes have neither cable television nor satellite dishes?
6. Farmer Louise owns 60 cows. She milks 35 percent of them a day. How many of the cows are not milked on a given day?
7. A family’s favorite cheese sauce uses a quarter-pound of cheddar, a quarter-pound of Muenster, and then equal amounts of five other cheeses to bring the total to a pound. Mozzarella is one of the other five cheeses. As a percentage, how much of the recipe is mozzarella?
8. An office building under construction will have 270 rooms, but only 30 percent are ready for occupancy. How many rooms are not ready?
9. Skip started out doing 40 jumping jacks a day. Now he can do 200 a day. As a percentage, how much has he increased his original daily total?
10. 60 percent of the 30 baseball teams each lost at least $10 million last year. How many teams fared better financially?
1. $750,000. Since we know that $900,000 is 50 percent larger than the 2000 fund total, determine the 2000 total by dividing $900,000 by 1.50. That is $600,000. The fund should be 25 percent larger than that; multiply $600,000 by 1.25. That is $750,000.
2. 20 percent. 300, the old figure, minus 240, the new figure, is 60. 60, the difference, divided by 300, the old figure, is .2. Multiply that by 100 and the answer is 20.
3. 5 percent. 483, the new figure, minus 460, the old figure, is 23. 23, the difference, divided by 460, the old figure, is .05. Multiply that by 100 and the answer is 5.
4. 2,875 miles and five days. This one is a little tricky, but at heart it’s really a percentage decrease problem. You need to find the “old” number (the total number of miles driven) and you know the “new” number (575 miles) and the percentage (20 percent).
There are at least two ways to solve the problem. Here’s the first one: You know that the “old” figure times .20 (the percentage expressed as a decimal) is 575. Since division is the reverse operation of multiplication, 575 divided by .20 will give you the “old” figure, 2,875 miles. Divide 2,875 by 575 and you get five days.
Divide 2,875 miles by 575 miles per day = five days
5. 10 percent
6. 39 cows
7. 10 percent
8. 189 rooms
9. 400 percent
10. 12 teams