**ratio**

A **ratio** is a comparison of two numbers.

*Example:*

3 to 5

What are the three ways to express a ratio of two numbers?

A ratio can be expressed in three different ways:

4 to 5

4 : 5

^{4}*/*_{5}

One of the ways to express a ratio is to write it as a fraction.

So, are ratios a lot like fractions?

Yes, ratios are a lot like fractions.

^{2}*/*_{3} = ^{4}*/*_{6} = ^{6}*/*_{9} = ^{200}*/*_{300} - these are equivalent fractions but also equal ratios.

The fraction doesn't change if both the numerator and the denominator are multiplied by the same number. So,

^{2}*/*_{3} = ^{4}*/*_{6} = ... = ^{2x}*/*_{3x} where *x* is a whole number.

How are ratios different from fractions?

A fraction shows part (the numerator) of the whole (the denominator).

A ratio can show a comparison between two parts of the same whole as well as a comparison between one part and the whole.

*Example*:

The ratio of cats to dogs at a pet shop is 2 to 3. At the very least, there are 5 (2 + 3) pets at this pet shop.

So, the ratio of cats to dogs is ^{2}*/*_{3}. The ratio of cats to all pets in the store is ^{2}*/*_{5}. The ratio of dogs to all pets in the store is ^{3}*/*_{5}.

**part-to-part** ratio

A **part-to-part** ratio represents a comparison between parts of a whole.

*Example*:

The ratio of girls to boys in a class is 3 to 5.

The ratio of plain bagels to poppy seed bagels to everything bagels is 5 : 4 : 3.

**part-to-whole** ratio

A **part-to-whole** ratio represents a comparison between one part of the ratio and a whole.

*Example*:

The ratio of boys to all students in the class is 3 to 8. The ratio of girls to all students in the class is 5 to 8.

How do you convert from a **part-to-part** ratio to a **part-to-whole** ratio?

A **part-to-part** ratio represents parts of a whole. To find the* "whole", add* the *"parts"* together.

*Example*:

The ratio of girls to boys in a class is 3 to 5. This is a part-to-part ratio. Add the "parts" to find the "whole". 3 + 5 = 8.

In other words, ^{3}/_{8} of all students are girls and ^{5}/_{8} of all students are boys.

Can you find the ratio of two numbers if you know the number of each item?

Of course, you can find the ratio if you know the actual number of each item.

*Example*:

There are 400 fiction books and 600 non-fiction books in a library. What is the ratio of fiction books to non-fiction books?

Reduce ^{400}*/*_{600} fraction to its lowest term. The ratio of fiction books to non-fiction books is 2 : 3.

What can you find if all you are given is the ratio of items?

You can find a number that the total number of items must be a *multiple* of.

*Example*:

The ratio of jeans to the tee shirts in your closet is 2 to 5.

^{2}*/*_{5} = ^{2x}*/*_{5x}. *x* could be any whole number. At the very least, there are 7 (2 + 5) jeans and tee shirts in your closet, but could also be 14 (4 + 10), 21 (6 + 15) or any multiple of 7.

If all you are given is a ratio of two items, can you find:

- the number of each item
- the total number of items?

No, you *cannot* find the number of each item or the total number of items.

*Example*:

The ratio of juniors to seniors at a party is 5 to 2. How many people are at the party? How many juniors? How many seniors?

All you can determine from the given is that the total number of people has to be a multiple of 7. The number of juniors has to be a multiple of 5 while the number of seniors has be a multiple of 2.

The ratio of juniors to seniors at a party is 5 to 2. There are 24 seniors at the party.

- Can you find the number of juniors?
- Can you find the total number of people at the party?

Yes, you can find both the number of juniors and the total number of people.

^{J}*/*_{S} = ^{5}*/*_{2} = ^{5x}*/*_{2x} = ^{?}*/*_{24 }

S = 24 = 2*x* ⇒ *x* = 12

J = 5 x 12 = 60

Total = J + S = 60 + 24 = 84

If you are given a ratio of items and the actual number of one part of the ratio, what can you determine?

You can determine the actual number of the other part of the ratio as well as the total number of items.

*Example*:

The ratio of boys to girls in your class is 5 to 3 and there are 18 girls.

^{5}/_{3} = ^{5x}/_{3x} = ^{?}/_{18}

3*x* = 18 ⇒ *x* = 6 ⇒ 5*x* = 30

30 + 18 = 48 - total number of students.

If you have two part-to-part ratios with a common part,

the ratio of* a *to* b* is 2 to 5

and

the ratio of* b *to* c *is 5 to 7,

what is the ratio of *a* to *c*?

The ratio of *a* to *c* is 2 to 7.

**rate**

A **rate** is a *ratio* between two measurements.

Normally, the two terms of a rate are measured in different units.

*Example: *

Miles per hour is the rate of speed.

What is the formula for the **rate of speed** of an object?

The** rate of speed** can be found by dividing distance by time.

**R = ^{D}/_{T} ⇒ D = R x T**

*Example:*

A car drives 100 miles in 2 hours. What is its rate of speed?

^{100}/_{2} = 50 mph

You walk from home to school with an average speed of 6 mph. On the way back from school your average speed is 4 mph.

What is your average speed for the entire trip?

The average speed for the entire trip is *not *(4+ 6) ÷ 2 = 5 mph! It equals 4.8 mph.

**Avg Speed = ^{Total Distance}/_{Total Time}**

D is the distance in *each* direction; therefore, the entire distance is 2D.

T1 is your time on the way from home to school. T2 is your time on the way back. The total time is the sum of T1 and T2.

T1 = ^{D}/_{S1}, T2 = ^{D}/_{S2}

**Avg Speed** = 2D*/*(T1 + T2) = 2D/(D/S1 + D/S2) = **2S1 * S2 / (S1 + S2)**

Avg Speed = 2 * 6 * 4*/*(6 + 4) = 4.8 mph

What is the formula for the **average rate** of speed?

**Average speed** is defined as total distance divided by total time.

**Average Speed = ^{Total Distance}/_{Total Time}**

**proportion**

Two *equal ratios* are called a **proportion**.

*a : b = c : d*

*Example:*

^{4}/_{7} = ^{12}/_{21}

How do you check if two ratios are equal?

^{1}/_{3} = ^{33}/_{99}

A common way to check if two ratios are equal is to *cross-multiply *the numbers making up those ratios.

^{1}*/*_{3} = ^{33}*/*_{99}

The product of 1 and 99 equals the product of 3 and 33; therefore, this proportion is a true proportion.

What method do you use to solve a proportion?

When solving a proportion, use the* cross-multiplication *method.

^{a}/_{b} = ^{c}/_{d}

*a * d = b * c*

*** Multiply the numerator of the fraction on the left side of the equation by the denominator of the fraction on the right side. Repeat for the other denominator and numerator.

When are two quantities in **direct proportion**?

Two quantities, A and B, are in **direct** **proportion **if both quantities change by the *same factor*.

*Example*:

1 can of soda costs $0.50. It would cost you $1.00 to buy 2 cans. For 6 cans you pay $3.00; for 12 cans you would pay $6.00. Notice that changing the number of cans you buy will change the total amount of money you pay. It changes by the same factor, 2.

If it takes 5 men 12 days to build a house, how many days would it take 6 men to build the same house if they work at the same rate?

Use simple logic to solve this problem.

Logically, the *more* men are on the job, the *less* time it's going to take for the job to be completed. This problem is an example of an indirect (or inverse) proportion.

A complete job would take 5 men * 12 days = 60 "men-days". Divide that total by the new number of workers. It would take 6 men *10 days* to build the same house.

Or you can set up a proportion with two ratios and take the reciprocal of one of them:

5 men → 12 days

6 men → *x* days

^{5}/_{6} = * ^{x}*/

_{12}⇒ 6

*x*= 60 ⇒

*x*= 10

When are two quantities in an **indirect (or inverse)** **proportion**?

Two quantities, A and B, are in **indirect **(**inverse)** **proportion** if by whatever *factor* A changes, B changes by a** ***reciprocal* of that factor.

*Example*:

When quantity A doubles, quantity B becomes half as large.

The terms exchange; the inverse of the ratio A/B is the ratio B/A.

What is the formula for indirect (inverse) proportions?

The formula for indirect proportions is:

*x _{1} : x_{2} = y_{2} : y_{1}*

Cross-multiply:

*x _{1} * y_{1} = x_{2} * y_{2}*

**percentage**

A **percentage **shows the ratio of a number to 100. It is denoted by the following symbol: %.

A **percentage **is a part of a whole (like a fraction or a decimal), expressed in hundredths.

*Example:*

2% = 0.02

What is 1%?

1% is the fraction ^{1}*/*_{100}.

*** "Per cent" literally means "per hundred".

What is *x*% of a number?

*x*% of a number is ^{x}*/*_{100} of that number.

How can you prove that *x*% of* y* always equals *y*% of *x*?

*x*% of *y* = ^{x}*/*_{100} * *y* = ^{xy}*/*_{100}

*y*% of *x* = ^{y}*/*_{100} * x = ^{yx}/_{100}

Two fractions are equal and, therefore,

*x% of y equals y% of x.*

How do you convert fractions into percentages?

*** Answer only for fractions whose denominators are *factors of 100*.

If the denominator of a fraction is a *factor* *of 100*, change the fraction to an equivalent fraction with a denominator of 100. The numerator of this equivalent fraction shows you the percentage.

*Example:*

^{3}*/*_{5} = ^{60}*/*_{100} = **60**%

How do you convert fractions into percentages?

*** Answer only for fractions whose denominators are *not factors of 100*.

If the denominator of a fraction is *not a factor of 100*, divide the numerator by the denominator, then multiply by 100.

*Example:*

^{4}/_{9} = 4 ÷ 9 = 0.4444 x 100 = 44.44%

^{1}*/*_{8} = 0.125 x 100 = 12.5%