RP Unit 1a:Ratios & Proportional

Reasoning

Greatest Common Factor• A factor is a number that you multiply by another number to get a product.

Example: 3 and 12 are factors of 36

Find the GCF of 32 and 60

1. List the factors of both numbers

32: 1 2 4 8 16 32

60: 1 2 3 4 5 6 10 15 20 30

2. Find the common factors

32: 1 2 4 8 16 32

60: 1 2 3 4 5 6 10 12 15 20 30 60

3. Select the greatest common factor . . . 4

Least Common Multiple• A multiple is a product of two factors. Example: 45 is a multiple of 5

Find the LCM of 6 and 8

1. List the first five multiples of both numbers

6: 6 12 18 24 30

8: 8 16 24 32 40

2. Find the least common multiple of the two numbers

6: 6 12 18 24 30

8: 8 16 24 32 40

3. If you haven’t found a common multiple, continue to find multiples of each number until you find one.

Ratio• A ratio is a comparison of two quantities.

• A ratio can be written three ways; using words, with a colon or as a fraction.

The ratio of dogs to cats is 2 to 3 or 2:3 or

• Order is important. The order of the numbers in your ratio must match the words. In the example above you couldn’t write your ratio 3 to 2.

Ratio• Ratios can be expressed in simplest form.

The ratio of roses to daisies is 4 to 12.

In simplest form: ÷ =

This means that for every 1 rose, there are 3 daisies.

• Ratios can compare parts to parts, parts to whole, or a whole to a part.

The ratio of daisies to all flowers is 4 to 16.

In simplest form the ratio of daisies to all flowers is 1 to 4.

This means that 1 out of every 4 flowers is a daisy.

Ratio• We can use ratios to divide something into equal groups.

George wants to divide his 42 baseball cards into two groups so that the ratio is 3 to 4.

1. Use a bar diagram to show a ratio of 3 to 4.

42

2. Divide the total by the number of equal sections to find out how many units are in each section

42 ÷ 7 = 6

Each section is equal to 6 units.

     

Ratio3. Since each unit is equal to 6, our ratio is 18 to 24

426 6 6

6 6 6 6

Rates• A rate is a ratio that compares two quantities with different kinds of units.

Example: miles per hour, price per pound, beats per minute

per means “for every”

I biked15 miles per hour means for every hour I biked, I traveled 15 miles.

Unit Rates• A unit rate is a rate where the second unit is 1.

Example: miles per hour means mile per 1 hour

12 gallons of water ran out of the faucet in 3 minutes. Write this rate as a unit rate. ÷ 3

12 gallons = ? gallons

3 minutes 1 minute

÷ 3

The water came out of the faucet at a rate of 4 gallons per minute.

Unit Prices• A unit price is the cost per unit.

Example: $3.99 per pound means each pound costs $3.99

6 pounds of chicken cost $18. What is the price per pound?

÷ 6

$18 = ? price

6 lb 1 lb

÷ 6

Each pound of chicken costs $3.

Use Unit Rates to Make Comparisons

Jim ran 21 laps of a track in 7 minutes. Ben ran 36 laps of the same track in 9 minutes. Which boy ran at a faster rate?

÷ 7 ÷ 9

Jim’s rate 21 laps= ? laps Ben’s rate 36 laps= ? laps

7 min 1 min 9 min 1 min

÷ 7 ÷ 9

Jim ran 3 laps in 1 minute. Ben rate 4 laps in 1 minute. Ben ran at a faster rate.

Ratio Tables

A ratio table has columns filled with pairs of numbers that have the same ratio.

A coed soccer team has a ratio of 4 boys for every 1 girl. If there are 16 boys on the team, how many players are girls?

 # of boys  4  8 12 16

# of girls 1 2 3 4

x 3

x 3x 4

x 4

x 2

x 2

Ratio Tables

You may need to scale back in order to scale forward in a ratio table.

To make strawberry jam you need 8 cups of sugar for every 14 cups of strawberries. How many cups of sugar would you need with 21 cups of strawberries?

There is no whole number by which you can multiply 8 to get 12.

 # of boys  8  4 12

# of girls 14 7 21

x 3

x 3

÷ 2

÷ 2

Ratios on the Coordinate PlaneCoordinate plane – a plane where a horizontal number (x-axis) intersects with a vertical number line (y-axis) at their zero points. The point of intersection is called the origin.

Ordered pair – a pair of numbers used to locate a point on the coordinate plane. The ordered pair is written as (x-coordinate, y-coordinate). The ordered pair for point A is (3, 5).

x-coordinate - the first number in an ordered pairy-coordinate - the second number in an ordered pair

Ratios on the Coordinate Plane

How many cups of chocolate chips do Sarah and Tyreak need to make 1 batch of cookies?

How many cups do they need to make 2 batches? 3 batches?

Ratios on the Coordinate PlaneMake a ratio table to show the ratio of the number of batches to cups of chocolate chips.

# of batches Cups of chips

0 0

1 2

2 4

3 6

4 8

5 10

6 12

Ratios on the Coordinate PlaneGraph the table.

# of batches, x

Cups of chips, y

Ordered pairs, (x, y)

0 0 (0, 0)

1 2 (1, 2)

2 4 (2, 4)

3 6 (3, 6)

4 8 (4, 8)

5 10 (5, 10)

6 12 (6, 12)

Describe the pattern. The graph shows that 2 cups of chocolate chipsare needed for each batch of cookies.

Equivalent RatiosTwo ways to determine if ratios or rates are equivalent are to use unit rates or multiply or divide one ratio to try to equal the other.

Find Unit Rates – find the unit rate of each ratio. If the unit rates are equal, the rates are equivalent.

20 cans = 5 cans 36 cans = 6 cans $4 $1 $6 $1Since 5 cans for $1 is not the same unit rate as 6 cans for $1, the rates are not equal.

Equivalent RatiosTwo ways to determine if ratios or rates are equivalent are to use unit rates or multiply or divide one ratio to try to equal the other.

Multiply or divide – if you can multiply (or divide) both terms of a ratio by the same number to equal the other ratio, the ratios are equal.Is 6 goals from 13 shots on goal equivalent to 12 goals from 23 shots on goal?

x 2

6 goals ? 12 goals

13 attempts = 23 attemptsx 2

6 x 2 = 12 but 13 x 2 ≠ 23 so the ratios are not equivalent.

Double Number LinesWe can use double number lines to solve ratio or rate problems.3 lbs of chicken costs $9. How much does 4 lbs of chicken cost at the same rate?

Step 1: Find the unit rate ÷ 3

Step 2: Multiply by the price for 1 lb x 4