Ratios
One thing most people overlook when planning to move to a new town is figuring out how many dangerous creatures live there. We won't make the same mistake, and luckily we have ratios to help us out.
A ratio is a way of comparing how much we have of two related things¬—say, vampires and werewolves.
Ratios can be written in three different ways. If your town has 25 vampires and 10 werewolves, the ratio of vampires to werewolves can be written as:
25 to 10
25:10
Notice that the last way looks a lot like a fraction. Ratios can even be reduced, exactly as we would reduce a fraction. By dividing both numbers by 5, that same ratio can be written as:
5 to 2
5:2
Proportions
When two ratios are equivalent, they can be written as a proportion. A proportion is two equal ratios. That doesn't always mean they have the same numbers in them. Here's what it does mean:
- If we simplify one ratio and simplify the other, they'll be equal.
- If we rewrite the ratios as decimals, they'll be equal.
- If the number of items in the first ratio goes up, the number of items in the second ratio will go up too.
- If a number is missing in one ratio, we can use the other ratio to figure it out.
Let's say the neighboring town has 5 vampires and 2 werewolves, so the ratio is 
. But how does their ratio compare to our town? When we simplify 
, we get 
, so we know the two towns have the same ratio of vampires to werewolves. We can write these ratios as a proportion.
In other words, a ratio of 25 vampires for every 10 werewolves is the same thing as 5 vampires for every 2 werewolves. We'd better rethink moving to this town.
Notice that the number of vampires is in both numerators, and the number of werewolves is in both denominators. Proportions always need to be set up like this, so that the numerators measure the same thing and the denominators measure the same thing. Keeping everyone in their proper place avoids epic battles of mythical creature power.
Ratios & Proportions Example 1
At a local bookstore, for every 100 books sold, 68 of them are fiction. What is the ratio of nonfiction books to total books sold? |
Ratios & Proportions Example 2
James has quite a bit of free time each day. He often spends 4 hours on the computer and the other 2 hours playing basketball. What is his ratio of basketball to computer time? What ratio of his total free time is spent on basketball? |
by dividing the numerator and denominator by 2.Ratios & Proportions Example 3
The ratio of frogs to tadpoles in a pond is 3:4. After scooping up some frogs and tadpoles in a jar, our ratio of frogs to tadpoles is 9:13. Is the ratio of amphibians in our jar proportional to the ratio of amphibians in the entire pond? |
by multiplying the top and bottom by 3. That gives us 
. And while 
is close to 
, it isn't equal, so these do not form a proportion. Ratios & Proportions Example 4
The ratio of glass beads to metal beads on a bracelet is 4:5, and the ratio of these beads on a necklace is 28:35. Do the bracelet and necklace have a proportional number of beads? |
. 
. 
. Now we have 
. Yep, they're proportional.Ratios & Proportions Exercise 1
Using the following table of movie ticket sales, what was the ratio of weekend gross sales for New Moon to The Blind Side?
Weekend Box Office Actuals (U.S.)
December 4-6, 2009
| Title | Distributor | Weekend Gross Sales | Cumulative Gross Sales | # of Theaters |
| The Blind Side | Warner Bros Pictures | $20,440,000 | $129,264,000 | 3326 |
| The Twilight Saga: New Moon | Summit Entertainment | $15,704,000 | $255,639,424 | 4124 |
| Brothers | Lionsgate Films | $9,700,000 | $9,700,000 | 2088 |
| Disney's A Christmas Carol | Walt Disney Pictures | $7,520,000 | $115,006,000 | 2546 |
| Old Dogs | Walt Disney Pictures | $6,901,000 | $33,934,000 | 3425 |
| Armored | Sony Pictures | $6,600,000 | $6,600,000 | 1915 |
| 2012 | Sony Picture | $6,600,000 | $148,787,000 | 3220 |
| Ninja Assassin | Warner Bros. Pictures | $5,030,000 | $29,790,000 | 2503 |
| Planet 51 | Sony Pictures | $4,300,000 | $33,966,000 | 2904 |
| Everybody's Fine | Miramax Films | $4,027,000 | $4,027,000 | 2112 |
Ratios & Proportions Exercise 2
Using the table from the first problem, what ratio of the top ten movies were distributed by Sony Pictures?
Ratios & Proportions Exercise 3
Using the table from the first problem, what ratio of 2012's cumulative gross sales was earned during the weekend of December 4 to December 6?
Ratios & Proportions Exercise 4
Using the table from the first problem, what was the ratio of movie theaters showing The Blind Side to theaters showing New Moon?