Download Extra “I`d rather have a fraction” Word Problems

E
L
P
M
Advanced
Fractions
A
S
“I’d rather have a fraction”
Multiplying, Cancelling, Dividing, Inverting
E
L
P
M
A
S
E
L
P
M
A
S
Copyright 2010 John McCormick All Rights Reserved
E
L
P
Advanced Fractions
“I’d Rather Have a Fraction”
Multiplying, Cancelling, Dividing, Inverting
M
A
S
E
L
P
M
A
S
Word Corner Publishing
15020 Burwood Drive
Lake Mathews, CA 92570
Advanced Fractions: Carlynn McCormick
Cheat Sheet: John McCormick
E
L
P
Illustrations: Microsoft Clipart
M
A
S
Copyright 2010
All Rights Reserved.
No part of this work may be copied
or duplicated in any form without
the express permission of the publisher.
E
L
P
Table of Contents
M
A
S
E
L
P
M
A
S
To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Multiplying, Cancelling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Dividing Inverting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
E
L
P
A Note About Retaining the Ability to do Fractions .
M
A
S
Fraction Cheat Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
17
To the Student
E
L
P
Purpose: This study guide has the purpose of helping student review advanced
fractions: how to cancel before multiplying fractions and how to invert when
M
A
S
dividing fractions to turn them into a multiplication problem. This guide has
attempted to make the review of advanced fractions as easy as eating fruit tarts.
E
L
P
M
A
S
Pre-requisite: You will want to complete the beginning fractions book: I’d rather
have a fraction, thank you before starting the advanced fraction book.
How to do the Study Guide: Do the steps in order and track your progress by
E
L
P
putting your initials and the date next to each step after completing it. Some steps
are done with your teacher or study partner. If you do not have a teacher or study
partner, you may do the study guide by yourself.
M
A
S
Important Note: To be competent in any subject you must know the vocabulary of
that subject. Be sure you know the meaning of all the words used in this study
guide.
Cancelling Before Multiplying Fractions
E
L
P
How to Take a Shortcut
M
A
S
E
L
P
M
A
S
Personally, I’d rather have shortcake than have a shortcut
READ: When it comes to multiplying fractions there is a shortcut that should be
E
L
P
taken. Before we learn the steps of the shortcut, let’s review some definitions:
Lowest Terms: When a fraction has been put in its simplest form possible.
Example: 4/6 is put in lowest terms as 2/3 (also called simplify or reduce).
M
A
S
Factor: A number that is being multiplied in a multiplication problem.
Example: In the problem 3 x 4 = 12, one factor is 3 and another factor is 4.
Common Factor: A number that is common to two different numbers when
those numbers are found by multiplying two numbers—one of which is the
common factor. Example: 2 is a common factor of the numbers 4 and 6 since
E
L
P
4 has the factors 2 x 2 and 6 has the factors 2 x 3.
Numerator: A fraction is made up of two numbers, one number on top and
another number on the bottom. The top number is the numerator. The
M
A
S
numerator tells how many of the parts are being counted.
Denominator: The bottom number is the denominator. The denominator
tells how many equal parts are in the whole. Example: 1/5 would require 5
parts to equal a whole.
Cancel: When you are multiplying fractions you should see if you can divide
the numerator of one fraction and the denominator of the other fraction by a
E
L
P
M
A
S
common number (one of the factors) which will give an answer that is
already in its simplest terms. If you don’t cancel before you multiply, then
your answer will need to be reduced to its simplest term. ____________
EXERCISE: Demonstrate the following to a study partner or your teacher: lowest
terms, factor, common factor, numerator, denominator and cancel. ____________
E
L
P
OPTIONAL: Make the definitions lowest terms, factor, common factor, numerator,
denominator and cancel in clay. ____________
M
A
S
READ: Now let’s look at the steps we take to get fractions in their lowest terms.
E
L
P
This is a shortcut that will make multiplying fractions quick and easy.
We will use 3/8 x 5/12 for the demonstration:
M
A
S
Step 1: Find a common factor of a numerator and the opposite denominator.
3 x 1 = 3 (numerator factors)
3 x 4 = 12 (denominator factors)
3 is a common factor
Step 2: Divide the numerator by that common factor. This is now the new
numerator of that fraction.
E
L
P
M
A
S
3÷3=1
Step 3: Divide the denominator by that common factor. This is now the new
denominator of that fraction.
12 ÷ 3 = 4
Step 4: If able, do steps 1-3 with the other numerator and opposite
denominator. You can now do the multiplication problem:
1/8 x 5/4 (newly canceled fractions)
Step 5: Multiply the 2 numerators to get the product’s numerator.
E
L
P
1 x 5 = 5 (numerator)
Step 6: Multiply the 2 denominators to get the product’s denominator.
M
A
S
8 x 4 = 32 (denominator)
This gives you the answer: 5/32
Note: If you don’t cancel the fractions before multiplying, the product of 3/8
x 5/12 would be 15/96. You would then need to reduce it to its lowest term
of 5/32.
EXERCISE: Work out how to cancel the fractions 5/12 x 4/15 using the six steps
E
L
P
above. You might see instantly that 5 and 15 have a common factor of 5 and that 4
and 12 have a common factor of 4. But go ahead and do the six steps, just to see how
they work.
M
A
S
READ: Cancelling is like working a puzzle. It is actually fun to do. You may not
need to do all six steps when problems are very easy. In the problem 6/8 x 2/9 you
might instantly know that that 6 and 9 have a common factor of 3; and that 2 and 8
have a common factor of 2. You would have the newly calculated fraction of 2/4 x
1/3 = 2/12 which can be reduced to 1/6. But even before you multiply the newly
E
L
P
M
A
S
calculated fraction, you might see that 2/4 can be reduced to 1/2 so you would
multiply 1/2 x 1/3 to get 1/6.
M
A
S
E
L
P
EXERCISE: For the following nine problems, cancel the fractions wherever possible
E
L
P
before multiplying them. Check your answers1 in the footnote below.
(1) 3/4 x 2/9 =
M
A
S
(2) 5/8 x 3/20 =
(3) 1/2 x 8/5 =
(4) 2/5 x 5/4 =
(5) 9/2 x 8/21 =
E
L
P
M
A
S
(6) 6/2 x 8/12 =
(7) 18/3 x 15/27 =
(8) 2/5 x 5/2 =
(9) 2/6 x 6/12 =
____________
M
A
S
E
L
P
I do not wish to reduce bonbons into fractions
1
Answers: (1) 1/6
(6) 4/2 = 2
(2) 3/32
(7) 10/3 = 3 1/3
(3) 4/5
(4) 1/2
(8) 1/1 = 1
(9) 1/6
(5) 1 5/7
EXERCISE: If you find cancelling and multiplying fun, then you will enjoy doing the
E
L
P
following exercise as well. Solve these problems by cancelling the fractions
wherever possible before multiplying them. Check your answers2 in the footnote
below.
M
A
S
(1) 11/25 x 5/7 =
(2) 23/52 x 43/ 46 =
(3) 20/23 x 7/2 =
(4) 11/3 x 12/10 =
E
L
P
M
A
S
(5) 3/4 x 2/3 =
(6) 150/144 x 12/15 =
_____________
E
L
P
When it comes to bonbons, I prefer improper fractions
2
Answers: (1) 11/35
(5) 1/2
M
A
S
(2) 43/104
(6) 10/12 = 5/6
(3) 70/23 = 3 1/23
(4) 44/10 = 4 2/5
Inverting Before Dividing Fractions
E
L
P
READ: When it comes to dividing fractions there are some tricks to learn. Before
we start, let’s review a definition and learn one:
M
A
S
Divide: To separate something into equal-sized parts. Example: You have a
pizza and six hungry people – you want to divide the pizza into sixths so each
person gets an equal share of the pizza.
Invert: To invert means to reverse the denominator and numerator of a
fraction. Example: If you invert 2/6, you get 6/2. If you invert 3/4, you get 4/3.
E
L
P
M
A
S
__________
EXERCISE: Show your teacher or study partner how easy it is to invert the fraction
1/2. Now have your teacher or study partner invert 5/8 and show you.
__________
READ: Here is another definition to learn:
Reciprocal: When two fractions are related to each other, such as 2/3 and
3/2, they are said to be reciprocal. When two reciprocal fractions are
multiplied together the answer is always 1. Example: 2/3 times 3/2 equals
E
L
P
6/6 or 1.
__________
M
A
S
EXERCISE: Draw out, make in clay or demonstrate the following to your teacher or
study partner: If you multiply 1/2 by its reciprocal 2/1 you get 2/2, which equals 1.
__________
EXERCISE: Explain to your teacher or study partner why multiplying the fraction
E
L
P
5/8 by its reciprocal will give you 1.
__________
READ: Here are the steps that tell you how to divide fractions: 3/7 ÷ 1/3 =
M
A
S
Step 1: Invert the fraction divisor (the divisor is the number you are
dividing by) to make a multiplication problem.
1/3 = 3/1 (invert divisor)
3/7 x 3/1 (division problem converted to multiplication problem)
E
L
P
M
A
S
Step 2: Multiply the numerators (top numbers):
3x3=9
Step 3: Multiply the denominators (bottom numbers):
7x1=7
3/7 x 3/1 = 9/7
Step 4: As needed, reduce the fraction to the lowest terms. If it is an
improper fraction change it to a mixed number: 9/7 = 1 2/7
E
L
P
__________
EXERCISE: Draw out, make in clay or demonstrate the four steps to your teacher or
M
A
S
study partner using the problem: 5/8 ÷ 2/3 = (by inverting the divisor and
multiplying, you will get the answer 15/16).
__________
READ: How do you divide if you have a mixed number such as 1 5/8 ÷ 2/3? You
E
L
P
simply change the mixed number into and improper fraction (8/8 + 5/8 = 13/8) and
divide: 13/8 ÷ 2/3; you get your answer by using the four steps above (which
include inverting and then multiplying).
M
A
S
__________
EXERCISE: Dividing fractions is similar to working a crossword puzzle—when you
know the guidelines and can figure out the answers—it is fun to do.
E
L
P
M
A
S
M
A
S
E
L
P
On the next page there are eight problems to use the four steps on. Remember if
you start out with a mixed number such as 4 9/18, you must change it to an
improper fraction: 81/18. (Here is the solution for getting the numerator: 4 x 18 =
72 + 9 = 81).
You can also reduce the fraction 81/18 – see problem 6 below. Check your
E
L
P
answers3 in the footnote below. If you get an improper fraction for an answer,
change it to a mixed number.
(1) 1/8 ÷ 1/3 =
M
A
S
(2) 3/4 ÷ 1/3 =
(3) 3/4 ÷ 5/7 =
E
L
P
M
A
S
(4) 3/7 ÷ 1/2 =
(5) 7/4 ÷ 21/16 =
(6) 4 9/18 ÷ 9/8 =
(Hint: 4 9/18 = 4 ½ = 9/2)
(7) 3 4/5 ÷ 2 29/31 =
(Hint: 3 4/5 = 19/5 and 2 29/31 = 91/31)
E
L
P
(8) 15/7 ÷ 2/4 =
3
Answers: (1) 3/8
(6) 4/1 = 4
M
A
S
(2) 9/4 = 2 ¼
(3) 21/20 = 1 1/20
(7) 589/455 = 1 134/455
(4) 6/7
(8) 60/40 = 4 4/14 = 4 2/7
__________
(5) 4/3 = 1 1/3