Numbers & Operations
Ratios
Math teachers love to use pizza slices to teach fractions and ratios. Who are we to argue with tradition? Consider a humble eight-slice pizza. You eat three pieces, and your imaginary buddy, Kronhorst, eats the other five. (Imaginary friends don’t have to worry about gaining weight.) The fraction of the pizza you ate can be determined by the part you ate over the whole number of slices. That’s what fractions are, a part to a whole.
Ratios compare parts to parts. The ratio of pieces you ate to the pieces Kronhorst ate is 3:5, because it compares the part you ate to the part Kronhorst ate. The ratio can be written as 3:5 or or ratio of 3 to 5. Even though looks like a fraction, it’s not. The bottom number is not a denominator.
If the ratio of A to B is 3:4, this does not necessarily mean that there are 3 pieces of A and 4 pieces of B. There could be 6 pieces of A and 8 pieces of B for a ratio of 6:8, which then reduces down to 3:4. Ratios don’t always tell you the actual amount, but they do allow you to compare one object to another:
2. For every 50 Americans who buy books online, 10 buy books at bookstores. What’s the ratio of those who buy books online to those who buy books at bookstores?
(A) 1:2
(B) 50:25
(C) 4:1
(D) 5:1
(E) 5:6
First find the ratio of those who buy books online to those who buy books at bookstores. This ratio can be written as 50:10.
You can simplify ratios the same way fractions are simplified:
50:10 = 5:1, choice D.
For every 5 people who buy books online, 1 person actually goes to the bookstore and buys a book. Whether that person actually reads it is another issue.
Proportions
Some items not only require knowledge of ratios but also test your ability to figure out the actual values from the ratios.
For instance, look at this item:
4. At a computer company in Brooklyn, the ratio of people who wear black, gray, and brown ties is 7:5:3. If the total number of workers is 45, how many workers wear gray ties?
(A) 3
(B) 5
(C) 9
(D) 15
(E) 21
You immediately know that for every 5 gray ties there are 7 black ones and 3 brown ones. You also know that for every 15 ties (7 + 5 + 3), 5 ties are gray and that the total number of ties is 45. Keep in mind that the ratios do not change, no matter the total number of objects. This helps you set up a proportion—an equation based on the notion that two ratios are equal.
5:15 is the same as x:45. To solve for x:
Now cross multiply to get:
455 = 15x
225 = 15x
x = 15
The total number of gray ties is 15, choice D.
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