Montgomery County Public Schools, Rockville, MD

[Pages:17]Answer Key

Sail into Summer with Math!

For Students Entering Investigations into Mathematics

This summer math booklet was developed to provide students in kindergarten through the eighth grade an opportunity to review grade level math objectives

and to improve math performance. THIS IS NOT A REQUIRED ASSIGNMENT

IM Summer Mathematics Packet

Table of Contents

Page

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Objective

Suggested Completion Date

Rename Fractions, Decimals, and Percents. . . . . . . . . . . . . June 22nd Fraction Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . June 25th Multiply Fractions and Solve Proportions . . . . . . . . . . . . . June 29th Add Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . July 6th

Subtract Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . July 9th Multiply Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . July 13th Divide Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . July 16th Decimal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . July 20th Find Percent of a Number . . . . . . . . . . . . . . . . . . . . . . . . . . July 23rd Solve Problems using Percent . . . . . . . . . . . . . . . . . . . . . . August 3rd Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . . . . . . . August 6th Integers I . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . August 13th Integers II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . August 17th Solving Equations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . August 20th Solving Equations II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .August 24th

Summer Mathematics Packet

Rename Fractions, Percents, and Decimals Hints/Guide:

3 To convert fractions into decimals, we start with a fraction, such as 5 , and divide the numerator (the top number of a fraction) by the denominator (the bottom number of a fraction). So:

6 5 | 3.0

- 30 0

3 and the fraction 5 is equivalent to the decimal 0.6

To convert a decimal to a percent, we multiply the decimal by 100 (percent means a ratio of a number compared to 100). A short-cut is sometimes used of moving the decimal point two places to the right (which is equivalent to multiplying a decimal by 100, so 0.6 x 100 = 60 and 3 5 = 0.6 = 60%

To convert a percent to a decimal, we divide the percent by 100, 60% ? 100 = 0.6 so 60% = 0.6

To convert a fraction into a percent, we can use a proportion to solve, 3x 5 = 100 , so 5x = 300 which means that x = 60 = 60%

Exercises:

No Calculators!

Rename each fraction as a decimal:

1 1. 5 =

0.2

3

2. 4 = 0.75

0.5 1

3. 2 =

0.8 1

8

4. 3 = 0.3333... 5. 10 =

Rename each fraction as a percent:

1

7. 5 = 20%

3

8. 4 = 75%

1

8

10. 3 = 33.33...% 11. 10 = 80%

Rename each percent as a decimal:

13. 8% = 0.08

14. 60% = 0.6

16. 12% = 0.12

17. 40% = 0.4

2

6. 3 = 0.6666....

1

9. 2 = 50%

2

12. 3 = 66.66....%

15. 11% = 0.11 18. 95% = 0.95

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Page 1

Summer Mathematics Packet

Hints/Guide:

Fraction Operations

When adding and subtracting fractions, we need to be sure that each fraction has the same

denominator, then add or subtract the numerators together. For example: 1 3 1 6 1+ 6 7 8+4=8+8 = 8 = 8

That was easy because it was easy to see what the new denominator should be, but what about if 78

it is not so apparent? For example: 12 + 15

For this example we must find the Lowest Common Denominator (LCM) for the two

denominators.

12 and 15

12 = 12, 24, 36, 48, 60, 72, 84, ....

15 = 15, 30, 45, 60, 75, 90, 105, .....

LCM (12, 15) = 60

So,

7 12

+

8 15

=

35 60

+

32 60

=

35 + 32 60

=

67 60

7 = 160

Note: Be sure answers are in lowest terms

To multiply fractions, we multiply the numerators together and the denominators together, and

then simplify the product. To divide fractions, we find the reciprocal of the second fraction (flip

the numerator and the denominator) and then multiply the two together. For example:

21 2 1

2 3 24 8

3 ? 4 = 12 = 6 and 3 ? 4 = 3 ? 3 = 9

Exercises: Perform the indicated operation:

No calculators!

SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page.

1 3 1. 4 + 5 =

6 2 2. 7 + 3 =

2 8 3. 5 + 9 =

3 2 4. 4 ! 3 =

2 2 5. 5 ! 9 =

9 2 6. 11 ! 5 =

1 2 7. 3 ? 3 =

3 3 8. 4 ? 5 =

7 2 9. 8 ? 5 =

3 3 10. 8 ? 4 =

1 1 11. 4 ? 4 =

7 3 12. 11 ? 5 =

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Page 2

Summer Mathematics Packet

Hints/Guide:

Multiply Fractions and Solve Proportions

To solve problems involving multiplying fractions and whole numbers, we must first place a one under the whole number, then multiply the numerators together and the denominators together. Then we simplify the answer:

6 6 4 24 3 7?4 = 7?1 = 7 = 37

To solve proportions, one method is to determine the multiplying factor of the two equal ratios.

For example:

4 24

4 24

9 = x since 4 is multiplied by 6 to get 24, we multiply 9 by 6, so 9 = 54 .

Since the numerator of the fraction on the right must be multiplied by 6 to get the numerator on the left, then we must multiply the denominator of 9 by 6 to get the missing denominator, which must be 54.

Exercises: Solve (For problems 8 - 15, solve for N):

No Calculators!

SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page.

3 1. 4 ? 4 =

1 2. 5 ? 7 =

1 3. 8 ? 5 =

3 4. 6 ? 7 =

4 5. 5 ? 4 =

2 6. 3 ? 6 =

1 7. 7 ? 4 =

1n 8. 5 = 20

3 12 9. n = 28

15 10. n = 25

n 12 13. 9 = 27

n 3 11. 4 = 12

2 18 14. 3 = n

3 12 12. 7 = n

2n 15. 7 = 21

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Page 3

Summer Mathematics Packet

Hints/Guide:

Add Mixed Numbers

When adding mixed numbers, we add the whole numbers and the fractions separately, then simplify the answer. For example:

1 8

4 3 = 4 24

6 18

+ 2 8 = 2 24

26

221

6 24 = 6 + 1 24 = 7 24 = 7 12

First, we convert the fractions to have the same denominator, then add the fractions and add the whole numbers. If needed, we then simplify the answer.

Exercises: Solve in lowest terms:

No Calculators!

SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page.

1

1.

2 4 1

+82

2.

3 185 1

+ 73

3

3.

3 5 1

+ 52

4.

3 5 8

1

+ 44

5.

3 7 7

1

+ 62

5 5 9 6. 1 + 13

1

7.

4 3 1

+ 64

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2

8.

1 3 1

+ 64

9.

2 1 9

2

+ 53

Page 4

Summer Mathematics Packet

Hints/Guide:

Subtract Mixed Numbers

When subtracting mixed numbers, we subtract the whole numbers and the fractions separately, then simplify the answer. For example:

3 18 7 4 = 7 24 15 15 - 2 24 = 2 24

3 1 5 24 = 5 8

First, we convert the fractions to have the same denominator, then subtract the fractions and subtract the whole numbers. If needed, we then simplify the answer.

Exercises: Solve in lowest terms:

No Calculators!

SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page.

1

1.

4 3 1

! 24

3

2.

6 4 2

! 3

2

3.

9 3 1

! 64

4.

3 6 4

1 ! 55

1

5.

7 2 1

! 34

6.

1 3 2

3 ! 210

1

7.

8 2 7

! 410

8.

1 8 3

5

! 56

5

9.

8 8 3

! 64

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Page 5

Summer Mathematics Packet

Hints/Guide:

Multiply Mixed Numbers

To multiply mixed numbers, we first convert the mixed numbers into improper fractions. This is

done by multiplying the denominator by the whole number part of the mixed number and then

adding the numerator to this product, and this is the numerator of the improper fraction. The

denominator of the improper fraction is the same as the denominator of the mixed number. For

example:

2

2 17

3 5 leads to 3 ? 5 + 2 = 17 so 3 5 = 5

Once the mixed numbers are converted into improper fractions, we multiply and simplify just as

with regular fractions. For example:

1 1 26 7 182 2 1 5 5 ? 3 2 = 5 ? 2 = 10 = 1810 = 18 5

Exercises: Solve and place your answer in lowest terms:

No Calculators!

SHOW ALL WORK. Use a separate sheet of paper (if necessary) and staple to this page.

11 1. 3 3 ? 4 2 =

21 2. 2 3 ?1 4 =

13 3.19 ? 4 5 =

31 4. 4 4 ?15 =

14 5. 3 3 ? 6 5 =

23 6. 6 3 ? 7 7 =

7.

1

4 5

?1

2 3

=

8.

2

2 5

?4

2 7

=

9.

4

1 3

?181

=

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Page 6

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