## What is a ratio?

#### How ratios are used

Ratios are used in both academic settings and in the real world to compare multiple amounts or quantities to each other. The simplest ratios compare only two values, but ratios comparing three or more values are also possible. In any situations in which two or more distinct numbers or quantities are being compared, ratios are applicable. By describing quantities in relation to each other, they explain how chemical formulas can be duplicated or recipes in the kitchen expanded. After you get to understand them, you will use ratios for the rest of your life.

#### What a ratio means

As noted above, ratios demonstrate the quantity of at least two items in relation to each other. So, for example, if a cake contains two cups of flour and one cup of sugar, you would say that the the ratio of flour to sugar was 2 to 1.

- Ratios can be used to show the relation between any quantities, even if one is not directly tied to the other (as they would be in a recipe). For example, if there are five girls and ten boys in a class, the ratio of girls to boys is 5 to 10. Neither quantity is dependent on or tied to the other, and would change if anyone left or new students came in. The ratio merely compares the quantities.

#### How are ratios expressed?

Ratios can be written out using words or can be represented using mathematical symbols.

- You will commonly see ratios represented using words (as above). Because they are used so commonly and in such a variety of ways, if you find yourself working outside of mathematic or scientific fields, this may the most common form of ratio you will see.
- Ratios are frequently expressed using a colon. When comparing two numbers in a ratio, you’ll use one colon (as in 7 : 13) and that when you’re comparing more than two numbers, you’ll put a colon between each set of numbers in succession (as in 10 : 2 : 23). In our classroom example, we might compare the number of boys to the number of girls with the ratio 5 girls : 10 boys. We can simply express the ratio as 5 : 10.
- Ratios are also sometimes expressed using fractional notation. In the case of the classroom, the 5 girls and 10 boys would be shown simply as 5/10. That said, it shouldn’t be read out loud the same as a fraction, and you need to keep in mind that the numbers do not represent a portion of a whole.

## Using Ratios

#### Reduce a ratio to its simplest form

Ratios can be reduced and simplified like fractions by removing any common factors of the terms in the ratio. To reduce a ratio, divide all the terms in the ratio by the common factors they share until no common factor exists. However, when doing this, it’s important not to lose sight of the original quantities that led to the ratio in the first place.

- In the classroom example above, 5 girls to 10 boys (5 : 10), both sides of the ratio have a factor of 5. Divide both sides by 5 (the greatest common factor) to get 1 girl to 2 boys (or 1 : 2). However, we should keep the original quantities in mind, even when using this reduced ratio. There are not 3 total students in the class, but 15. The reduced ratio just compares the relationship between the number of boys and girls. There are 2 boys for every girl, not exactly 2 boys and 1 girl.
- Some ratios cannot be reduced. For example, 3 : 56 cannot be reduced because the two numbers share no common factors – 3 is a prime number, and 56 is not divisible by 3.

#### Use multiplication or division to “scale” ratios

One common type of problem that employs ratios may involve using ratios to scale up or down the two numbers in proportion to each other. Multiplying or dividing all terms in a ratio by the same number creates a ratio with the same proportions as the original, so, to scale your ratio, multiply or divide through the ratio by the scaling factor.

- For example, a baker needs to triple the size of a cake recipe. If the normal ratio of flour to sugar is 2 to 1 (2 : 1), then both numbers must be increased by a factor of three. The appropriate quantities for the recipe are now 6 cups of flour to 3 cups of sugar (6 : 3).
- The same process can be reversed. If the baker needed only one-half of the normal recipe, both quantities could be multiplied by 1/2 (or divided by two). The result would be 1 cup of flour to 1/2 (0.5) cup of sugar.

#### Find unknown variables when given two equivalent ratios

Another common type of problem that incorporates ratios asks you to find an unknown variable in one ratio, given the other number in that ratio and a second ratio that is equivalent to the first. The principle of cross multiplication makes solving these problems fairly simple. Write each ratio in its fractional form, then set the two ratios equal to each other and cross multiply to solve.

- For example, let’s say we have a small group of students containing 2 boys and 5 girls. If we were to maintain this proportion of boys to girls, how many boys would be in a class that contained 20 girls? To solve, first, let’s make two ratios, one with our unknown variables: 2 boys : 5 girls = x boys : 20 girls. If we convert these ratios to their fraction forms, we get 2/5 and x/20. If you cross multiply, you are left with 5x=40, and you can solve by dividing both figures by 5. The final solution is x=8.

## Catching Mistakes

#### Avoid addition or subtraction in ratio word problems.

Many word problems look something like this: “A recipe calls for 4 potatoes and 5 carrots. If you want to use 8 potatoes instead, how many carrots will you need to keep the ratio the same?” Many students try to add the same amount of each quantity. You actually need to use multiplication, not addition, to keep the ratio the same. Here’s an example of the wrong and right to solve this example:

- Wrong method: “8 – 4 = 4, so I added 4 potatoes to the recipe. That means I should take the 5 carrots and add 4 to that too… wait! That’s not how ratios work. I’ll try again.”
- Right method: “8 ÷ 4 = 2, so I multiplied the number of potatoes by 2. That means I should multiply the 5 carrots by 2 as well. 5 x 2 = 10, so I want 10 carrots total in the new recipe.”

#### Convert to the same units

Some word problems get tricky by switching to a different unit partway through. Convert to the same unit before finding the ratio. Here’s an example problem and solution:

- A dragon has 500 grams of gold and 10 kilograms of silver. What is the ratio of gold to silver in the dragon’s hoard?
- Grams and kilograms are not the same unit, so we’ll need to convert. 1 kilogram = 1,000 grams, so 10 kilograms = 10 kilograms x 1,000grams1kilogram {\displaystyle {\frac {1,000grams}{1kilogram}}} = 10 x 1,000 grams = 10,000 grams.
- The dragon has 500 grams of gold and 10,000 grams of silver.
- The ratio of gold to silver is 500gramsGold10,000gramsSilver=5100=120 {\displaystyle {\frac {500gramsGold}{10,000gramsSilver}}={\frac {5}{100}}={\frac {1}{20}}}.

#### Write your units in the problem

In ratio word problems, it’s much easier to catch mistakes if you write the units after each value. Remember, the same unit on the top and bottom of a fraction cancels out. After you cancel out as much as you can, you should end up with the right units for your answer.

- Example problem: If you have six boxes, and in every three boxes there are nine marbles, how many marbles do you have?
- Wrong method: 6boxes∗3boxes9marbles=…{\displaystyle 6boxes*{\frac {3boxes}{9marbles}}=…}Wait, nothing cancels out, so my answer would be “boxes x boxes / marbles.” That doesn’t make sense.
- Right method:

6boxes∗9marbles3boxes={\displaystyle 6boxes*{\frac {9marbles}{3boxes}}=}6boxes∗3marbles1box={\displaystyle 6boxes*{\frac {3marbles}{1box}}=}6boxes∗3marbles1box={\displaystyle {\frac {6boxes*3marbles}{1box}}=}6∗3marbles1={\displaystyle {\frac {6*3marbles}{1}}=}18 marbles.

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