Fractions and Decimals

This Math Pak is the property of

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THE MATH PAK

Fractions and Decimals

INSTRUCTIONS
Welcome to Math Team!

This package contains fraction and decimal arithmetic exercises. If you complete all these exercises, you will be well on your way to Math Team! Here's what we want you to do:

Work each unit in order.
Work as many problems are you can in as many units as you can. You don't have to work all the problems in every unit. Just do your best!
No calculators! Please don't use a calculator to work these problems.
Check your answers with the answers in the answer page.
If you missed any, try to figure out where you made your mistakes. If you still can't figure it out, ask your parents, or get someone to help you. Don't erase! Just correct your problem over the old answer.
Turn in your completed Math Pak (complete with corrected problems) when Math Team begins!

Yay! You are on your way!

INTRODUCTION TO FRACTIONS

Fractions are expressed as one number over another number, like this:

\(\huge {1 \over 2} \leftarrow \text{This means a half of something}\)
The number on the top is called the numerator and the number on the bottom is called the denominator.

When you think of a fraction, think of a PIZZA!!

Suppose a pizza is cut evenly into the number of pieces in the DENOMINATOR.
If the number of pieces YOU get is the NUMERATOR, the fraction of the pizza you get is:

Adding and taking away (subtracting) fractions can be pictured using slices of pizza. For example:

Multiplying fractions means cutting a portion into smaller portions.
Example 1: \(\Large {1 \over 2} \times {1 \over 2} = {1 \over 4}\) Means: a half of a half is a fourth	Example 2: \(\Large {1 \over 2} \times {1 \over 3} = {1 \over 6}\) Means: a half of a third is a sixth

Unit 1 show you how to do this multiplication.

INTRODUCTION TO FRACTIONS (CONTINUED)

DIVIDING FRACTIONS

Dividing fractions means determining how many smaller pieces there are in a larger piece.

\(\Large {1 \over 2} \div {1 \over 4} = \normalsize 2\)

This means one-half divided by one-forth is 2.
In other words, there are 2 one-forth pieces of pizza in a half pizza.

Unit 4 shows you how to do this division.

Here's another way to look at fractions. Imagine that you have a board and you want to cut it into halves, thirds, fourths, sixths, and fifths. Here is what those board fractions look like:

<===================one whole board!!====================>

¹⁄₂ board ¹⁄₂ board

¹⁄₄ board ¹⁄₄ board ¹⁄₄ board ¹⁄₄ board

¹⁄₃ board ¹⁄₃ board ¹⁄₃ board

¹⁄₆ board ¹⁄₆ board ¹⁄₆ board ¹⁄₆ board ¹⁄₆ board ¹⁄₆ board

¹⁄₅ board ¹⁄₅ board ¹⁄₅ board ¹⁄₅ board ¹⁄₅ board

Notice that the 2 ¹⁄₄th boards add up to a half board and that the 2 ¹⁄₆ boards add up to a ¹⁄₃ board.

This means that there are 2 one-fourth boards in a half-board and that there are two one-sixth boards in a one-third board!

So now, let's begin!

Name:_________________Parent:____________________Points:_________

UNIT 1: FRACTION MULTIPLICATION

To multiply fractions, you multiply the numerators together and then multiply the denominators together. The numerator is the number on top and the denominator is the number on the bottom. The result (the "product") is always smaller than either of the original fractions.

EXAMPLES:
\(\Large {1 \over 2} \times {1 \over 2} = {1 \times 1 \over 2 \times 2} = {1 \over 4}\)

"Half of a half is one fourth"
\(\Large {1 \over 8} \times {3 \over 4} = {1 \times 3 \over 8 \times 4} = {3 \over 32}\)
"three quarters of an eighth
is three thirty-seconds".

Multiply these fractions: (I will get you started with the first one)

1). \(\Large {1 \over 2} \times {1 \over 4} = {1 \times 1 \over 2 \times 4} = \) 2). \(\Large {1 \over 3} \times {1 \over 3} = \)

3). \(\Large {1 \over 3} \times {1 \over 2} = \)

4). \(\Large {1 \over 5} \times {1 \over 4} = \)

5). \(\Large {1 \over 6} \times {1 \over 2} = \) 6). \(\Large {1 \over 6} \times {1 \over 6} = \)

7). \(\Large {1 \over 9} \times {1 \over 9} = \)

8). \(\Large {1 \over 9} \times {1 \over 8} = \) 9). \(\Large {5 \over 8} \times {7 \over 9} = \)

10). \(\Large {3 \over 11} \times {1 \over 4} = \)

11). \(\Large {3 \over 5} \times {1 \over 5} = \) 12). \(\Large {1 \over 6} \times {1 \over 8} = \)

13). \(\Large {1 \over 8} \times {1 \over 2} = \)

14). \(\Large {1 \over 4} \times {1 \over 4} = \) 15). \(\Large {1 \over 4} \times {1 \over 8} = \)

16). \(\Large {1 \over 8} \times {3 \over 4} = \)

17). \(\Large {1 \over 4} \times {3 \over 4} = \) 18). \(\Large {3 \over 8} \times {1 \over 8} = \)

19). \(\Large {3 \over 5} \times {1 \over 4} = \)

20). \(\Large {2 \over 7} \times {1 \over 7} = \) 21). \(\Large {3 \over 8} \times {1 \over 4} = \)

22). \(\Large {1 \over 5} \times {1 \over 5} = \)

23). \(\Large {1 \over 12} \times {1 \over 2} = \) 24). \(\Large {1 \over 3} \times {1 \over 5} = \)

25). \(\Large {8 \over 9} \times {1 \over 3} = \)

26). \(\Large {7 \over 8} \times {1 \over 4} = \) 27). \(\Large {5 \over 6} \times {5 \over 6} = \)

28). \(\Large {4 \over 5} \times {4 \over 5} = \)

29). \(\Large {5 \over 9} \times {1 \over 3} = \)
The following problems are hard!

30). \(\Large {1 \over 12} \times {1 \over 12} = \) 31). \(\Large {1 \over 50} \times {1 \over 10} = \) 32). \(\Large {126 \over 254} \times {50 \over 61} = \)

Name:_________________Parent:____________________Points:_________

UNIT 2: FRACTION ADDITION - SAME DENOMINATORS

To add fractions that have the same denominators, you just add the numerators (the top numbers). You keep the same denominator.
You DON'T add the denominators!

EXAMPLES:

\(\Large {1 \over 4} +{1 \over 4} = {1 +1 \over 4} = {2 \over 4}\) \(\Large \leftarrow\) Don't add the denominators! \(\Large \rightarrow\) \(\Large {1 \over 5} +{3 \over 5} = {1 +3 \over 5} = {4 \over 5}\)

Add these fractions: (I will get you started with the first one)

1). \(\Large {1 \over 18} +{1 \over 18} = {\text{ } \over 18}\) 2). \(\Large {3 \over 5} +{2 \over 5} = \)

3). \(\Large {1 \over 3} +{1 \over 3} = \)

4). \(\Large {1 \over 8} +{2 \over 8} = \)

5). \(\Large {1 \over 9} +{1 \over 9} = \) 6). \(\Large {1 \over 16} +{1 \over 16} = \)

7). \(\Large {1 \over 25} +{1 \over 25} = \)

8). \(\Large {1 \over 36} +{4 \over 36} = \) 9). \(\Large {1 \over 27} +{3 \over 27} = \)

10). \(\Large {1 \over 56} +{2 \over 56} = \)

11). \(\Large {1 \over 21} +{3 \over 21} = \) 12). \(\Large {1 \over 18} +{6 \over 18} = \)

13). \(\Large {1 \over 5} +{1 \over 5} = \)

14). \(\Large {3 \over 8} +{4 \over 8} = \) 15). \(\Large {1 \over 7} +{3 \over 7} = \)

16). \(\Large {3 \over 11} +{6 \over 11} = \)

17). \(\Large {4 \over 6} +{1 \over 6} = \) 18). \(\Large {7 \over 81} +{6 \over 81} = \)

19). \(\Large {16 \over 21} +{1 \over 21} = \)

20). \(\Large {5 \over 61} +{8 \over 61} = \) 21). \(\Large {2 \over 23} +{11 \over 23} = \)

22). \(\Large {18 \over 23} +{2 \over 23} = \)

23). \(\Large {1 \over 17} +{8 \over 17} = \) 24). \(\Large {6 \over 15} +{5 \over 15} = \)

25). \(\Large {6 \over 17} +{4 \over 17} = \)

26). \(\Large {9 \over 28} +{10 \over 28} = \) 27). \(\Large {4 \over 91} +{5 \over 91} = \)

28). \(\Large {9 \over 11} +{2 \over 11} = \)

29). \(\Large {1 \over 2} +{1 \over 2} = \)
The following problems are hard!

30). \(\Large {126 \over 251} +{126 \over 251} = \) 31). \(\Large {342 \over 831} +{288 \over 831} = \) 32). \(\Large {1251 \over 2555} +{899 \over 2555} = \)

Name:_________________Parent:____________________Points:_________

UNIT 3: FRACTION SUBTRACTION - SAME DENOMINATORS

Fraction subtraction is similar to fraction addition. If the denominators are the same, just subtract the second numerator from the first one. The denominator stays the same.
You DON'T subtract the denominators!
EXAMPLES:

\(\Large {3 \over 4} - {1 \over 4} = {3 -1 \over 4} = {2 \over 4}\) \(\Large \leftarrow\) Don't subtract the denominators! \(\Large \rightarrow\) \(\Large {4 \over 5} -{2 \over 5} = {4 -2 \over 5} = {2 \over 5}\)

Subtract these fractions:

1). \(\Large {4 \over 5} -{1 \over 5} = {4-1\over5} = {\text{ } \over 5}\) 2). \(\Large {7 \over 8} -{1 \over 8} = \)

3). \(\Large {5 \over 6} -{1 \over 6} = \)

4). \(\Large {3 \over 7} -{1 \over 7} = \)

5). \(\Large {5 \over 8} -{3 \over 8} = \) 6). \(\Large {7 \over 9} -{2 \over 9} = \)

7). \(\Large {9 \over 10} -{1 \over 10} = \)

8). \(\Large {9 \over 11} -{1 \over 11} = \) 9). \(\Large {15 \over 16} -{1 \over 16} = \)

10). \(\Large {9 \over 11} -{7 \over 11} = \)

11). \(\Large {15 \over 16} -{11 \over 16} = \) 12). \(\Large {11 \over 13} -{9 \over 13} = \)

13). \(\Large {8 \over 9} -{7 \over 9} = \)

14). \(\Large {5 \over 7} -{4 \over 7} = \) 15). \(\Large {6 \over 7} -{4 \over 7} = \)

16). \(\Large {14 \over 21} -{9 \over 21} = \)

17). \(\Large {55 \over 60} -{50 \over 60} = \) 18). \(\Large {11 \over 23} -{8 \over 23} = \)

19). \(\Large {17 \over 21} -{13 \over 21} = \)

20). \(\Large {27 \over 32} -{22 \over 32} = \) 21). \(\Large {81 \over 90} -{80 \over 90} = \)

22). \(\Large {44 \over 56} -{22 \over 56} = \)

23). \(\Large {14 \over 15} -{11 \over 15} = \) 24). \(\Large {13 \over 56} -{9 \over 56} = \)

25). \(\Large {21 \over 23} -{11 \over 23} = \)

26). \(\Large {18 \over 31} -{9 \over 31} = \) 27). \(\Large {11 \over 14} -{9 \over 14} = \)

28). \(\Large {58 \over 51} -{26 \over 51} = \)

29). \(\Large {77 \over 78} -{58 \over 78} = \)
The following problems are hard!

30). \(\Large {126 \over 255} -{88 \over 255} = \) 31). \(\Large {811 \over 812} -{188 \over 812} = \) 32). \(\Large {1241 \over 2560} -{899 \over 2560} = \)

Name:_________________Parent:____________________Points:_________

UNIT 4: FRACTION DIVISION

To divide one fraction by another, you merely turn the second fraction upside-down (invert it) and then multiply! EXAMPLES:

Sometimes, fraction division problems look like this:
\(\Large {({1 \over 8}) \over ({1 \over 4})} = \Large {1 \over 8} \div {1 \over 4} = {1 \over 8} \times {4 \over 1} = {4 \over 8}\)

Divide these fractions: (I will invert the first one) Rewrite the problem as a multiply first!

1). \(\Large {1 \over 2} \div {1 \over 4} = {1 \over 2} \times {4 \over 1} = \)

2). \(\Large {1 \over 2} \div {3 \over 4} = \)

3). \(\Large {1 \over 3} \div {1 \over 9} = \)

4). \(\Large {1 \over 3} \div {2 \over 9} = \)

5). \(\Large {1 \over 5} \div {1 \over 4} = \)

6). \(\Large {1 \over 2} \div {1 \over 9} = \)

7). \(\Large {1 \over 4} \div {1 \over 8} = \)

8). \(\Large {1 \over 4} \div { 8 \over 1} = \)

9). \(\Large {3 \over 4} \div {8 \over 1} = \)

10). \(\Large {1 \over 4} \div \large 8 = \)

11). \(\Large {1 \over 2} \div {1 \over 16} = \)

12). \(\Large {3 \over 4} \div {1 \over 16} = \)

13). \(\Large {\text{1/5} \over \text{2/10}} = \)

14). \(\Large {2 \over 5} \div {1 \over 10} = \)

15). \(\Large {3 \over 5} \div {1 \over 10} = \)

16). \(\Large {1 \over 4} \div {8 \over 12} = \)

17). \(\Large {1 \over 5} \div \normalsize 10 = \)

18). \(\Large {3 \over 8} \div {3 \over 1} = \)

19). \(\Large {3 \over 8} \div \normalsize 3 = \)

20). \(\Large {\text{3/8} \over \text{8/5}} = \)

21). \(\Large {3 \over 8} \div {1 \over 4} = \) 22). This problem is hard!

\(\Large {16 \over 31} \div {8 \over 81} = \)

Name:_________________Parent:____________________Points:_________

UNIT 5: CHANGING DENOMINATORS

To change the denominator of a fraction to another number, divide the denominator you want by the denominator you have and then multiply by your numerator to get the new numerator.

EXAMPLES:

\(\Large {1 \over 2} = {\text{ ? } \over 8}\) \(\leftarrow\) ?=4 because (8/2) x 1 =4 \(\Large {3 \over 4} = {\text{ ? } \over 8} \) \(\leftarrow\) ?=6 because (8/4) x 3 = 6

Change these fractions to their new denominators:

1). \(\Large {1 \over 2} ={ \text{ } \over 4}\) (Hint: divide 4 by 2)	2). \(\Large {1 \over 3} ={\text{ } \over 9} \)	3). \(\Large {2 \over 3} ={\text{ } \over 9} \)
4). \(\Large {1 \over 4} ={\text{ } \over 8} \)	5). \(\Large {3 \over 4} ={\text{ } \over 8} \)	6). \(\Large {1 \over 8} ={\text{ } \over 16} \)
7). \(\Large {6 \over 8} ={\text{ } \over 16} \)	8). \(\Large {5 \over 6} ={\text{ } \over 12} \)	9). \(\Large {1 \over 8} ={\text{ } \over 32} \)
10). \(\Large {1 \over 4} ={\text{ } \over 16} \)	11). \(\Large {1 \over 9} ={\text{ } \over 18} \)	12). \(\Large {2 \over 9} ={\text{ } \over 18} \)
13). \(\Large {5 \over 8} ={\text{ } \over 16} \)	14). \(\Large {9 \over 15} ={\text{ } \over 30} \)	15). \(\Large {4 \over 15} ={\text{ } \over 30} \)
16). \(\Large {9 \over 15} ={\text{ } \over 30} \)	17). \(\Large {1 \over 32} ={\text{ } \over 64} \)	18). \(\Large {12 \over 18} ={\text{ } \over 36} \)
19). \(\Large {11 \over 44} ={\text{ } \over 88} \)	20). \(\Large {9 \over 11} ={\text{ } \over 44} \)	21). \(\Large {5 \over 16} ={\text{ } \over 32} \)
22). \(\Large {15 \over 31} ={\text{ } \over 62} \)	23). \(\Large {8 \over 9} ={\text{ } \over 81} \)	24). \(\Large {5 \over 7} ={\text{ } \over 49} \)
25). \(\Large {8 \over 16} ={\text{ } \over 64} \)	26). This problem is hard! \(\Large {8 \over 211} ={\text{ } \over 422} \)

Name:_________________Parent:____________________Points:_________

UNIT 6: FRACTION ADDITION - DIFFERENT DENOMINATORS (PART 1)

To add fractions with different denominators, you must first change both denominators to be the same. If one denominator divides evenly into the other, then the new denominator will be the LARGER of the 2 denominators.

EXAMPLES:

Add these fractions: (I will give you the first 2 common denominators)

1). \(\Large {3 \over 4} +{1 \over 2} = {3 \over 4} + {\text{ } \over 4} = \)

2). \(\Large {1 \over 3} +{1 \over 9} = {\text{ } \over 9 } + {1 \over 9} = \)

3). \(\Large {2 \over 3} +{1 \over 9} = \)

4). \(\Large {1 \over 4} +{1 \over 8} = \)

5). \(\Large {1 \over 4} +{3 \over 8} = \)

6). \(\Large {1 \over 4} +{2 \over 8} = \)

7). \(\Large {3 \over 5} +{2 \over 10} = \)

8). \(\Large {1 \over 16} +{1 \over 32} = \)

9). \(\Large {3 \over 4} +{1 \over 8} = \)

10). \(\Large {1 \over 5} +{1 \over 10} = \)

11). \(\Large {1 \over 5} +{3 \over 10} = \)

12). \(\Large {1 \over 4} +{5 \over 16} = \)

13). \(\Large {1 \over 4} +{7 \over 16} = \)

14). \(\Large {3 \over 4} +{3 \over 16} = \)

15). \(\Large {3 \over 4} +{5 \over 16} = \)

16). \(\Large {1 \over 16} +{9 \over 32} = \)

17). \(\Large {1 \over 16} +{3 \over 32} = \)

18). \(\Large {1 \over 4} +{1 \over 16} = \)

19). \(\Large {3 \over 4} +{1 \over 16} = \)

20). \(\Large {3 \over 8} +{1 \over 64} = \)

These 2 problems are hard!

21). \(\Large {3 \over 9} +{5 \over 81} = \) 22). \(\Large {126 \over 256} +{9 \over 512} = \)

Name:_________________Parent:____________________Points:_________

UNIT 7: FRACTION ADDITION - DIFFERENT DENOMINATORS (PART 2)

To add fractions with different denominators, you must first change both denominators to be the same. If one denominator DOES NOT divide evenly into the other, then make the new denominator the product (multiply) of the 2 denominators.

\(\Large {1\over2} + {1\over3}\qquad\qquad\leftarrow\) EXAMPLES:\(\Large\rightarrow \qquad\qquad {1\over4} + {1\over3}\)

new denomiator = \(2 \times 3 = 6\)
new denomiator = \(3 \times 4 = 12\)

Add these fractions: (I will give you the first 2 common denominators)

1). \(\Large {1 \over 4} +{1 \over 6} = {\text{ } \over 24} + {\text{ } \over 24} = \)	2). \(\Large {3 \over 4} +{1 \over 3} = {\text{ } \over 12 } + {\text{ } \over 12} = \)
3). \(\Large {1 \over 4} +{1 \over 5} = \)	4). \(\Large {1 \over 4} +{2 \over 5} = \)
5). \(\Large {1 \over 4} +{3 \over 5} = \)	6). \(\Large {3 \over 4} +{3 \over 6} = \)
7). \(\Large {3 \over 5} +{2 \over 3} = \)	8). \(\Large {3 \over 4} +{2 \over 5} = \)
9). \(\Large {3 \over 4} +{3 \over 5} = \)	10). \(\Large {3 \over 4} +{4 \over 5} = \)
11). \(\Large {1 \over 5} +{1 \over 6} = \)	12). \(\Large {1 \over 5} +{2 \over 6} = \)
13). \(\Large {2 \over 5} +{1 \over 6} = \)	14). \(\Large {3 \over 5} +{1 \over 6} = \)
15). \(\Large {3 \over 4} +{1 \over 5} = \)	16). \(\Large {4 \over 5} +{1 \over 6} = \)
17). \(\Large {1 \over 5} +{5 \over 6} = \)	18). \(\Large {1 \over 8} +{1 \over 7} = \)
19). \(\Large {1 \over 8} +{3 \over 7} = \)	20). \(\Large {5 \over 6} +{6 \over 8} = \)

21). \(\Large {8 \over 11} +{5 \over 8} = \)

\(\large\leftarrow\) These 2 problems\(\large\rightarrow\)
are hard!

22). \(\Large {15 \over 13} +{8 \over 11} = \)

Name:_________________Parent:____________________Points:_________

UNIT 8: FACTORING

To 'factor' (a verb) a number means to break it up into numbers that can be multiplied together to get the original number. The numbers that you break it into are called factors (noun!) (Just a little confusing!)
EXAMPLES:

\( 6 = 3 \times 2 \leftarrow\) the factors of 6 are 3 and 2

\( 9 = 3 \times 3 \leftarrow\) the factors of 9 are 3 and 3 Sometimes numbers can be factored in different combinations:
\( 8 = (4 \times 2)\) or \( (2 \times 2 \times 2)\)

\( 18 = (9 \times 2) \) or \( (6 \times 3)\) or \( (2 \times 3 \times 3)\)

Factor the following numbers. I will factor the first one for you!

1). \( 10 = 5 \times 2\)	2). \( 15 = \)	3). \( 20 = \)
4). \( 21 = \)	5). \( 22 = \)	6). \( 4 = \)
7). \( 12 = \)	8). \( 14 = \)	9). \( 26 = \)
10). \( 27 = \)	11). \( 25 = \)	12). \( 30 = \)
13). \( 28 = \)	14). \( 50 = \)	15). \( 33 = \)
16). \( 34 = \)	17). \( 70 = \)	18). \( 46 = \)
19). \( 39 = \)	These next 3 problems are hard!
20). \( 169 = \)	21). \( 95 = \)	22). \( 221 = \)

Name:_________________Parent:____________________Points:_________

UNIT 9: REDUCING TO LOWEST TERMS

If the numerator and denominator of a fraction contain the same factor, they can be reduced to lowest terms by removing that factor. EXAMPLES:

\(\Large \require{cancel} {2 \over 4} = {{1 \times 2} \over {2 \times 2}} = {{1 \times \cancel{2}} \over {2 \times \cancel{2}}} = \Large {1 \over 2}\)
Numerator and denominator both
have a factor of 2 \(\Large \require{cancel} {6 \over 9} = {{2 \times 3} \over {3 \times 3}} = \Large {{2 \times \cancel{3}} \over {3 \times \cancel{3}}} = \Large {2 \over 3}\)
Numerator and denominator both
have a factor of 3

Reduce these fractions to their lowest terms:
(I will give you the common factors for the first 3 problems!)

1). \(\Large {10 \over 15} = {(2 \times 5)\over (3 \times 5)} = \)	2). \(\Large {6 \over 12} = \) (common factor is 6)	3). \(\Large {4 \over 8} = \) (common factor is 4)
4). \(\Large {9 \over 27} = \)	5). \(\Large {12 \over 18} = \)	6). \(\Large {6 \over 8} = \)
7). \(\Large {6 \over 10} = \)	8). \(\Large {6 \over 14} = \)	9). \(\Large {6 \over 18} = \)
10). \(\Large {6 \over 20} = \)	11). \(\Large {8 \over 16} = \)	12). \(\Large {3 \over 9} = \)
13). \(\Large {5 \over 20} = \)	14). \(\Large {5 \over 25} = \)	15). \(\Large {5 \over 30} = \)
16). \(\Large {10 \over 12} = \)	17). \(\Large {10 \over 14} = \)	18). \(\Large {12 \over 14} = \)
19). \(\Large {8 \over 14} = \)	20). \(\Large {7 \over 14} = \)	21). \(\Large {6 \over 16} = \)
22). \(\Large {3 \over 15} = \)	23). \(\Large {5 \over 15} = \)	24). \(\Large {18 \over 27} = \)
25). \(\Large {18 \over 36} = \)	26). This problem is hard! \(\Large {55 \over 66} = \)

Name:_________________Parent:____________________Points:_________

UNIT 10: FRACTIONS - MIXED NUMBERS

A fraction in which the numerator is larger than the denominator is called an improper fraction.
These are improper fractions:

\(\Large {10 \over 8}\) \(\Large {12 \over 6}\) \(\Large {15 \over 9}\) \(\Large {21 \over 10}\)
Fractions like these can be turned into mixed numbers.
A mixed number is a whole number with a fraction added to it, like \(\large 1 \Large{1\over 2}\).
You turn an improper fraction into a mixed number by dividing the numerator by the denominator and making a fraction that goes with it (if there is one) by putting the remainder over the denominator.
The above improper fractions change into mixed numbers by dividing the numerator by the denominator, like this:

Turn these improper fractions into mixed numbers:

1). \(\Large {20 \over 8} = \)
Hint: 8 goes into 20 how many times? ___
what is the remainder? ___

2). \(\Large {6 \over 4} =\)

3). \(\Large {6 \over 5} =\)

4). \(\Large {12 \over 8} =\)	5). \(\Large {12 \over 9} =\)	6). \(\Large {12 \over 10} =\)
7). \(\Large {12 \over 11} =\)	8). \(\Large {18 \over 6} =\)	9). \(\Large {18 \over 7} =\)
10). \(\Large {18 \over 8} =\)	11). \(\Large {18 \over 9} =\)	12). \(\Large {25 \over 12} =\)
13). \(\Large {25 \over 13} =\)	14). \(\Large {7 \over 4} =\)	15). \(\Large {7 \over 3} =\)
16). \(\Large {7 \over 2} =\)	17). This problem is hard! \(\Large {50 \over 8} =\)	18). This problem is hard! \(\Large {250 \over 9} =\)

Name:_________________Parent:____________________Points:_________

UNIT 11: FRACTIONS - MULTIPLYING MIXED NUMBERS

To multiply or divide mixed numbers by any other number (fraction, whole number or another mixed number) you must first convert the mixed number to an improper fraction.

You can add or subtract with mixed numbers but you can't multiply or divide with them!
To convert a mixed number to an improper fraction, you multiply the denominator by the whole number and then add the numerator to get your new numerator. The denominator stays the same. This is just the opposite of what we did in unit 10.

EXAMPLES:

\( 5\Large {1 \over 8} = {{8\times5+1} \over 8} = {41 \over 8}\) \( 2\Large {3 \over 5} = {{5\times2+3} \over 5} = {13 \over 5}\) \( 4\Large {1 \over 3}={{3\times4+1} \over 3} = {13 \over 3}\)
Here's how to think about this:
- Take the first mixed number: \( 5\Large {1\over8}\)
- How many 8ths are there in 5? There are 40 of them. (Each whole number has 8 8ths.)
- Add 1 to get \(\Large {41\over 8}\)

Warning:
It doesn't work to just multiply the whole numbers and the fractions separately!

\( 2\Large {1\over2} \times \normalsize 2\Large{1\over2}\) is not \( (2 \times 2) + \Large ({1\over2}\times{1\over2})\normalsize = 4 + \Large {1\over4} = \normalsize 4\Large{1\over4}\leftarrow\) Wrong answer!
(See problem #8)

Multiply the following: (I will convert the first mixed number for you!)
You may leave your answer as a improper fraction, but you get an extra point (each!) for converting your answer back to a mixed number!

1). \( 1\Large{1\over3}\times \normalsize 2 = \Large{4\over 3} \times {2\over 1} = \) Note: the 2 becomes \(\large {2\over1}\) as a fraction	2). \( 3\Large{1\over8}\times{1\over2} = {\text{ }\over 8} \times {1\over 2} = \)
3). \( 2\Large{1\over6}\times{1\over3} = \)	4). \( 3\Large{2\over3}\times{1\over4} = \)
5). \( 6\Large{1\over5}\times{1\over3} = \)	6). \( 4\Large{1\over3}\times \normalsize 2 \Large {1\over2} = \)
7). \( 6\Large {4\over5} \times \normalsize 1\Large{1\over2} = \)	8). \( 2\Large {1\over2} \times \normalsize 2\Large{1\over2} = \)
9). \( 2\times 3\Large {1\over2} = \)	10). \( 4\Large{1\over8}\times \normalsize 4 \Large{1\over8} = \)
11). This problem is hard! \(\rightarrow 15\Large{1\over2}\times \normalsize 15\Large{1\over2} = \)

Name:_________________Parent:____________________Points:_________

UNIT 12: INTRODUCTION TO DECIMALS

Decimals are numbers that contain a decimal point, like:

\( 1.5\) \( 2.8\) \( 0.2\) \( 3.14\) \( 66.66\)

These are numbers that fall between the integers.
For example, \(1.5\) is between 1 and 2. \(66.66\) is between 66 and 67.

The number to the left of the decimal point is an integer (positive or negative whole number or zero).
The number to the right of the decimal is called the "tenths" digit and tells you the number of tenths (\( \Large {1\over10}\)ths) that are added to the whole number to the left of the decimal point. So...

1.5 is the same as the mixed number \(1\Large{5\over10}\) 2.8 is \(2\Large{8\over10}\) and 0.2 = \(\Large {2\over10}\)

If there are 2 numbers to the right of the decimal point then both numbers to the right of the decimal point are the number of "hundredths" (\(\Large {1\over100}\)ths) to be added to the whole number, so.....

3.14 is the same
as the mixed number
\( 3\Large{14\over100}\)

66.66 is \(66 \Large {66\over100}\)

and 0.25 = \(\Large {25\over100}\)

Convert these decimals to mixed numbers:

1). \( 1.6 = \)	2). \( 2.9 = \)	3). \( 14.4 = \)
4). \( 16.17 = \)	5). \( 22.88 = \)	6).This problem is hard! \( 1.04 = \)

Convert these mixed numbers to decimals:

7). \( 4 \Large{ 6\over10} = \)	8). \( 2 \Large{ 3\over10} = \)	9). \( 6 \Large{ 8\over10} = \)
10). \( 9 \Large{ 56\over100} = \)	11). \( 15 \Large{ 88\over100} = \)	12). This problem is hard! \( 88 \Large{ 5\over100} = \)

Name:_________________Parent:____________________Points:_________

UNIT 13: DECIMAL ADDITION AND SUBTRACTION

To add or subtract decimals, you must write one above the other with their decimal points lined up, like this:

\( 1.5\)
\( \underline{ +2.6}\)

   \( 8.95\)
\( \underline{+1.9}\)

\( 44.11\)
\(-\) \( \underline{0.2}\)

\( 14.\)
\(-\) \(\underline{ .04}\)
Where one decimal doesn't have digits that the other one does, you put in zeroes, like this:

\( 1.5\)
\( \underline{ +2.6}\)

\( 8.95\)
\( \underline{+1.90}\)

   \( 44.11\)
\(-\) \( \underline{00.20}\)

\( 14.00\)
\(-\) \(\underline{ 00.04}\)
Now, its easy! You just add or subtract the two numbers as you would two whole numbers, with carries or borrows if necessary, but you keep track of where the decimal point goes and put it in the result right under the decimal points in the two numbers being added or subtracted , like this:

\( 1.5\)
\( \underline{ +2.6}\)
   \(4.1\)
\( 8.95\)
\( \underline{ +1.90}\)
\(10.85\)
\( 44.11\)
\(-\) \( \underline{00.20}\)
\(43.91\)
\( 14.00\)
\(-\) \(\underline{ 00.04}\)
\(13.96\)

Add or subtract these decimals: (I will line up the first decimal for you!)

1). \( 2.2\) \( \underline{ +1.1}\)	2). \( 4.4 + 8.8\)	3). \( 9.09 + .2\)
4). \( 16. + 1.05\)	5). \( 19.09 + .02\)	6). \( 16.33 + 8.25\)
7). \( 9.09 + .2\)	8). \( 16. - 1.05\)	9). \( 19.09 - .02\)
10). \( 16.33 - 8.25\)	11). \( 26.55 - 1.2\)	12). \( 99.99 - .001\)
13). This problem is hard! \( 100. - .099\)	14). This problem is hard! \( 9.99 + 100.99\)

Name: _______________________________ Parent:______________________ Points: ________

UNIT 14: DECIMAL MULTIPLICATION

To multiply 2 decimals, you merely multiply the 2 numbers together, ignoring the decimal points.
Then, when you have your answer (called the "product"), you add the number of digits to the right of the decimal point in each number together and place the decimal in your answer (product) that many places from the RIGHT.

YOU DON'T LINE UP THE DECIMAL POINTS!

EXAMPLES:

\( \quad\;\, 1.2 \quad \leftarrow \;\,\text{ 1 digit to the right of the decimal point}\)
\(\;\; \times \underline{2.4} \quad\leftarrow + \underline{1} \text{ digit to the right of the decimal point}\)
\(\quad \;\,4\,8 \qquad\quad\, \text{ 2 digits to the right in the product}\)
\(\;\;\, \underline{2\,4}\)
\(\;\;\, 2.88 \)
\(\quad \uparrow\)
\(\text{ 2 digits to the right in the product}\)

\( \quad\;\, 3.11 \quad \leftarrow \;\,\text{ 2 digits to the right}\)
\(\quad \times \underline{2.4} \quad\leftarrow + \underline{1} \text{ digit to the right}\)
\(\quad 1244 \qquad\quad\, \text{ 3 digits to the right in the product}\)
\(\quad \underline{622}\)
\(\;\;\, 7.464 \)
\(\quad \uparrow\)
\(\text{ 3 digits to the right in the product}\)

Note: You should always do a 'reasonableness check' when multiplying decimals to verify that your answer is reasonable. In the second example, above, just ignore the numbers past the decimals and multiply \(3 \times 2 = 6\). The answer should be close to 6. The answer, 7.464 passes this reasonableness check. It is close to 6!

Multiply these decimals (don't use your calculator and DON'T LINE UP THE DECIMAL POINTS!):

1). \(3.6\) \(\times \underline{ 1.3}\)	2). \(18.2\) \(\times \underline{ 1.1}\)	3). \(1.2\) \(\times \underline{ 9.}\)
4). \(8.6\) \(\times \underline{ 4.4}\)	5). \(11.22\) \(\times \underline{ .4}\)	6). \(4.44\) \(\times \underline{ 1.6}\)
7). \(.44\) \(\times \underline{ 6.}\)	8). \(.49\) \(\times \underline{ .06}\)	9). \(.12\) \(\times \underline{ .06}\)

Name: _______________________________ Parent:______________________ Points: ________

UNIT 15: DECIMAL DIVISION

Here's how you divide one decimal by another: Suppose you want to divide 4.5 by 1.8
First, you write your problem as you normally would: Next, you move the decimals in both the numbers to the right the same number of times until the divisor is a whole number. The decimal point in your answer will be lined up with the one in the number you are dividing into. Now, you do your division: Next, if you have a remainder, bring down a zero from the number being divided and continue: Check your answer for reasonableness by ignoring the part past the decimal and dividing in your head! For example, in this last problem 45/18 is about 2 so the answer should be near 2.

Divide these decimals: (These problems are all hard but don't use your calculator!)

1).

\( 1.1 \text{ } \overline{)\text{ }5.5\text{ }} = \)

2).

\( 2.4 \text{ } \overline{)\text{ }8.4\text{ }} = \)

3).

\( 4.0 \text{ } \overline{)\text{ }5.0\text{ }} = \)

4).

\( 2.0 \text{ } \overline{)\text{ }15.5\text{ }} = \)

5).

\( .04 \text{ } \overline{)\text{ }1.88\text{ }} = \)

6).

\( 8.8 \text{ } \overline{)\text{ }.440\text{ }} = \)

Congratulations! You have finished!