UNIT 1: Ratios, Rates, & Proportions

UNIT 1: Ratios, Rates, & Proportions

Review: fractions A fraction allows you to determine two quantities and their proportion to each other as part of a whole. NUMERATOR number on top (part) DENOMINATOR number on bottom (whole) 2 10 For example: You ate four slices of pizza. The pizza was cut into 8 slices. You ate 4 slices of pizza. The pizza was cut into 8 slices. The slices that you ate is represented by 4. 4 parts of the whole 8 8 eaten.

Review: decimals A decimal is a part of a whole number broken down over increments of 10 (exponentially 10, 100, 1000, 10000, etc.) For example: 4 slices of pizza of out 8 have been eaten. This represents eaten 4 of the pizza eaten. 8 Use your calculator, what decimal do you get? 4 8 =.5

REVIEW: decimals

Ratios A ratio allows you to compare two quantities that are of the same nature, in a given order and have the same type of units. You can compare for example: Boys with boys Girls with girls Boys with girls apples with oranges circles with squares Length of a rectangle with the width of a rectangle and much more!!!

Ratios The ratio of the number a to the number b is represented as: a:b or a b a is the first term of the ratio and b is the second term of the ratio. Find the ratio in this example: For every 2 steps Billy takes he walks 5 meters. The first term is 2 The second term is 5 The ratio 2:5

Terms in the Ratio A term is a number that represents a quantity in a ratio. Ex. 5 to 8 4 : 7 9 2 Terms

Find the ratio What is the ratio of: a) All apples to oranges b) Green apples : All Fruits 6 to 3 3:9 c) all apples all fruits 6 9

Part-to-part ratios A Part-to-Part ratio is when you compare 2 different parts of a group to each other. The ratio of circles to squares 4 to 6 circles : squares circles squares 4:6 4 6

Part-to-whole ratios A Part-to-Whole ratio is when you compare part of the group to the whole group. The ratio of circles to all shapes can be represented in 3 ways: 4 : 10 or 4 to 10 or 4 10 The ratio of squares to all shapes can be represented in 3 ways: 6 : 10 or 6 to 10 or 6 10

Practice ratios Christine is 3 years old and weighs 20 kg. Mr. Freckles is 50 years old and weighs 77 kg. 1) What is the ratio of Christine s age to Mr. Freckles age? 2) What is the ratio of Mr. Freckles mass to Christine s mass? ***Write the ratios in all three ways.

Reducing ratios METHOD 1: 24 36 Remove the GCF (Greatest Common Factor) from the Numerator and Denominator. Then divide each term or the ratio by the GCF. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 24 36 12 = 12 = 2 3

Reducing ratios METHOD 2: Divide the Numerator and Denominator by the same number until you cannot anymore. 40 120 2 = 2 = 10 = 20 60 10 30 2 = 2 = 10 = 1 3

Your turn to reduce a ratio Reduce 24: 108 or 24 108 2 2 3 24 108 12 = = = 54 6 27 2 9 2 2 3

Scale factor (SF) The scale factor is the number you multiply or divide each term in the ration to get the equivalent terms in another ratio.

What is the scale factor (SF)? 10 : 12 = 30 : 36 SF = 3 4 to 10 = 16 to 40 SF = 4 7 8 = 35 40 SF = 5

Rate A rate is a comparison of two quantities measured in different types of units. *** Unlike ratios, rates always include units. _25 km_ or _1 photocopy_ or _12 apples_ 1 hr 10 $ 4

Speed is a very common rate Speed is measured by distance over time Ex. _100 meters_ = ( 10) 10 meters = 10 seconds ( 10) 1 second 10 m/s

Rate example For Example: Peter earned $27 for 3 hours worth of work. How much did he make per hour (Unit = money & Unit = hour)? $27 3h = $9 per hour or 3h $27 =.11 hours per dollar (18min to earn $1)

Equivalent rates An equivalent rate is a rate that represents the same comparison. _12 apples_ = _3 apples_ $ 4 $1 _100 km_ = _50 km 2 h 1 h

Unit rate Unit rates are rates that have a second unit (or denominator) that is 1. Ex. _2 laps_ or _3 apples_ 1 minute $1 or _100 km = 100 km/h 1 hour

How to convert to a unit rate 1) Write your rate as a fraction (include units!!!) 2) Divide your numerator by your denominator, this is now your new numerator. 3) Your denominator becomes 1. Ex. 400 km in 3.5 hours 400 km 3.5 hours 400 3.5 = 114.2857. 114.29 114.29 km 1 hour

Practice: converting to unit rate You just got your first summer job! On the first day, you worked 7 hours and made 120 dollars. How much did you make per hour? (This is unit rate!)

Comparison of Ratios and of Rates Two methods to compare ratios and rates: 1) We find the same common denominator or the same basis of comparison. Ex: 2 3 < 5 7 14, since < 15 21 21 70 words 4 min > 95 words 6 min, since 210 words 12 min > 190 words 12 min

Practice method 1 comparing rates and ratios

Comparison of Ratios and of Rates 2) We calculate their quotients (divide the numerator by the denominator). Ex: 50 > 60 20 25, since 50 20 = 2.5 and 60 25 = 2.4 600 m 5s < 500 m 4 s, since 120 m/s < 125 m/s

Practice method 2 comparing rates and ratios

Comparing rates It is very important to know how to compare rates! So how do you do this? Simple! Change your rates to unit rates Ex. Which store charges more for pears? Store A: $12 for 10 pears Store B: $8 for 6 pears

Important conversions : time Hours to minutes to seconds X 60 X 60 1 hr = 60 minutes = 3600 seconds 60 minutes 1 hour 60 seconds 1 minute A) How many hours is 240 minutes? B) How many minutes is 480 seconds?

Important conversions : distance kilometers to meters meters to kilometers X 1000 1000 1 km = 1000 m 1000 m = 1 km 1 km 1000 m A) How many meters is 2.5 kilometers? B) How many kilometers is 550 meters?

Practice: comparing rates Which car is traveling faster? Car A: 100 km / hour Car B: 2 hours to travel 220 km Car C: 40 km in 30 minutes

Equivalent ratios and rates If two ratios or rates have the same quotient (result once divided), they are equivalent. Ex: 1) the ratios 8 5 and 24 15 are equivalent since 8 5 = 1.6 and 24 15 = 1.6 2) the rates 10 km 1 hr and 40 km 4 hrs are equivalent since 10 km 1 hrs = 10 km/hr and 40 km 4 hrs = 10 km/hr

Extremes Proportions When two ratios or rates are compared and determined to be equivalent, they are proportional. Proportions involves comparing, multiplying and dividing. So! If the ratio of a to b is equal to the ratio of b to c then a:b = c:d or a b = c d is a proportion. A proportion has 4 terms: Means a: b = c:d

Proportion: property of the crossproducts In every proportion, the product of the end terms is equal to the product of the middle terms. So, from the proportion a b = c d, we find the equality ad = bc Cross-products a b = c d From the proportion 4 2 = 16 8,we know that 4 x 8 = 2 x 16 32 = 32

Proportions practice Are the following proportions true or false? a) 12 = 10 18 15 b) 21 = 24 49 56 c) 1.4 = 3.4 3.5 8.5 d) 2.5 = 0.5 7.5 2.5

A new proportion given a proportion Given a proportion, we get a new proportion by - interchanging the end terms if a b = c d, then d b = c a - interchanging the middle terms if a b = c d, then a c = b d - inverting the ratios if a b = c d, then b a = d c

A new proportion given a proportion Given the proportion 5 3 = 10 6, find three new proportions. 1) 2) 3)

Finding a missing term in a proportion Use cross-products! Ex: 3 = 15 24 x 3 = 15 24 x 15 times 24 divided by 3 15 24 3 = x 120 = x

Find the missing proportion An employee s salary is proportional to the duration of his work. For 3 hours of work, he receives $24 and for 12 hours, $96. How much will de receive for 15 hours of work? Number of hours 3 12 15 Salary ($) 24 96 x HINT! Use cross-products!

Proportions: challenge question 7 pounds of strawberries makes 8 fruit roll-ups. If every batch of fruit roll-up requires the same amount of strawberries, how many pounds of strawberries are required to make 20 fruit rollups? A) 0.875 B) 2.5 C) 17.5 D) 23 *** HINT: Set up a ratio and use cross-product!!!