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Contents page Topic Pg 3 Whole Numbers

Multiplication square Exercise 1

Pg 4 Mathematical Terminology Pg 8 Number Properties Exercise 2 Pg 9 Calculator work Exercise 3 Pg 10 Number Properties Exercise 4 Pg 11 Number Properties Exercise 5 Pg 13 BODMAS Pg 14 BODMAS Exercise 6 Pg 15 Vertical method for addition and

subtraction Exercise 7

Pg 16 Horizontal method for addition and subtraction

Exercise 8

Pg 17 Horizontal method for multiplication Exercise 9 Pg 18 Vertical method for multiplication Exercise 10 Pg 19 Long Division Exercise 11 Pg 19 Calculator work Exercise 12 Pg 20 Ratio & Rate Exercise 13 Pg 21 Calculating Ratio Exercise 14 Pg 22 Ratio & Rate Exercise 15 Pg 23 Calculating percentage Exercise 16 Pg 24 VAT, Interest and discount Pg 25 Calculating Profit & Loss Exercise 17

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GRADE SEVEN MATHEMATICS

TOPIC 1: WHOLE NUMBERS

Exercise 1: Mental calculations

You can do calculations more easily and quickly if you know the multiplication tables up to

“12 times”; division up to 144; and all your addition/subtraction bonds.

Copy the following table in your informal book, then complete: (Suggestion: copy the table

in one colour, and the answers in another.)

X 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

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TERMINOLOGY Our number system consists of the following:

1. Whole numbers This includes all numbers from 0 to infinity.

The symbol used is No.

2. Natural numbers This includes all numbers from 1 to infinity.

The symbol used is N.

3. Prime numbers A prime number has only 2 factors: 1 and itself.

Example: F3 = {1; 3} 3 is therefore a prime number.

4. Composite numbers A composite number has more than 2 factors.

Example: F20 = {1; 2; 4; 5; 10; 20}

5. Factors Factors are those numbers that can fit equally into another number

without any remainders. F10 = {1; 2; 5; 10}

6. Prime factors A prime factor is a factor that is also a prime number.

7. Multiples Multiples are those numbers obtained when multiplying the number

Example: M7 = {7; 14; 21; 28; 35; ……..}

8. Ordering numbers

Smallest to biggest – Ascending

Biggest to smallest – Descending

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9. Comparing Numbers Use symbols: < means “less than”

: > means “greater than”

10. Round off numbers to the nearest 5, 10, 100 or 1000 Here is a reminder of how to round off a number to the nearest 10:

Underline the Tens digit .....586

Look at the digit to the RIGHT of the Tens digit....586

If this digit is 0, 1, 2, 3, or 4, the Tens stay the same If this digit is 5, 6, 7, 8 or 9, round up 586 rounded to the nearest 10 is 590.

We use the same method to round off to 100 and 1000.

11. Inverse Operations Multiplication and division are inverse operations. They can be used

to check answers. Example: 2+4 = 4+2; 3x6 = 6x3

12. Properties of Numbers

A. COMMUTATIVE PROPERTY The commutative property changes the order of numbers.

2 + 4 = 4 + 2

9 x 6 = 6 x 9

(What conclusion can you draw about the commutative property?)

Explain what you understand by commutative property.

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THE ASSOCIATIVE PROPERTY The associative property changes the way numbers are grouped,

but still keeps the numbers in the given order.

Study the examples given below:

(20 + 10) + 15 = 20 + (10 + 15)

(5 x 20) x 4 = 5 x (20 x 4)

What conclusion can you draw about the associative property?

B. DISTRIBUTIVE PROPERTY The distributive property means multiplication is applied to addition

and subtraction to make the calculation easier to work out.

Study the following examples:

2 x 5 + 6 - 7 123 x 7

= 2 x (5 + 6 - 7) =(100 + 20 + 3) x 7

=2 x 4 =(100 x 7)+(20 x 7)+(3 x 7)

= 8 = 700 + 140 + 21

= 861

C. RE-ARRANGEMENT PROPERTY The rearrangement property re-arranges or changes the given

order of the numbers and groups like operations together to make

the sum easier to calculate.

Study the examples below:

15 + 13 – 7 + 74 – 3 1 000 ÷ 8 x 6 ÷3 x 2

= 15 + 13 + 74 – 7 – 3 = 1 000 x 6 x 2 ÷ 8 ÷ 3

7

= 102 – 7 – 3 = 12 000 ÷ 8 ÷ 3

= 92 = 500

D. THE GROUPING OF SUBTRACTION AND DIVISION METHODS (SUCCESSIVE SUBTRACTION AND SUCCESSIVE DIVISION) 65 – 12 - 13 – 27 32 400 ÷ 25÷2 ÷ 2

= 65 – (12 + 13 + 27) = 32 400÷(25x2x 2)

= 65 – 52 = 32 400 ÷ 100

= 13 = 324

13. When multiplying or dividing by multiples of 10, 100, 1000 54 x 300

= 54 x 3 x 100 (“Break down” the one number)

= 162 x 100

= 16200.

14. Identity element for addition It is 0.

Whether you add or subtract 0 from a number, the number never

changes

15. Identity element for multiplication It is 1

Whether you multiply or divide a number by 1, the number never

changes

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Exercise 2 1. Why is the number 1 neither a prime nor a composite number?

2. Write down the following sets of numbers:

a) Odd numbers between 100 and 120.

b) Even numbers from 364 to 372.

c) Prime numbers greater than 5 but smaller than 27.

d) The first 5 multiples of 12.

e) Multiples of 8 from 48 to 80.

f) The first 10 composite numbers.

g) The factors of 144.

h) The first 5 counting numbers.

i) The first 5 natural numbers.

j) The prime factors of 30.

3. Determine if the second number is a multiple of the first number.

Give a reason for your answer.

a) 16; 8

b) 36; 48

c) 25; 125

d) 100; 10

e) 20; 200

4. Write down ALL the factors of the following numbers:

a) 50

b) 25

c) 48

5. Write down the first 6 multiples of the following:

a) 10

b) 25

c) 125

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Exercise 3 YOU MAY USE YOUR CALCULATOR IN THIS EXERCISE 1. Solve the problems below by first rounding off each number to the nearest

10 000 and then calculating the actual answer.

a. 171 643 + 16 124

170 000 + 20 000

190 000

b. 399 106 + 71 257 + 9 199

2. Repeat the same method as above, but this time round off the nearest whole number.

a. 128,69 – 99,6

b. 34,9056 ÷ 5,4

3. Use the same method again to round off these numbers to the nearest 100: a. 9 876 543 – 210 369

b. 12 413 x 125

4. Round off the following numbers to the nearest 10: a. 8 342 x 29

b. 211 x 43

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Exercise 4

COMPLETE THE FOLLOWING: 1. USE THE REARRANGEMENT PRINCIPLE

a) 72 – 37 + 15 + 7 – 28

b) 36 ÷ 9 X 5 ÷ 4

c) 753 + 124 – 16 – 109 + 312

d) 1 000 ÷ 8 X 6 ÷ 3

e) 40 ÷ 10 ÷ 20 X 10

2. USE THE GROUPING OF SUBTRACTION AND DIVISION NUMBERS (SUCCESSIVE SUBTRACTION AND SUCCESSIVE DIVISION) a) 100 – 13 – 36 – 45

b) 1 000 ÷ 2 ÷ 5 ÷ 10

c) 1 860 – 120 – 384 – 239

d) 12 300 ÷ 100 ÷ 10

e) 136 598 – 3 750 – 2 146 – 789

3. THE DISTRIBUTIVE PROPERTY a) 29 x 4 + 29 X 7 + 29 x 10 =

b) 116 + 203+ 290=609

c) 58 x 19= 1102

d) 795 x 6= 4770

e) 124 x 50 = 6200

f) 450 ÷ 10 + 680 ÷ 10 =45 + 68 = 113

g) 1 250 ÷ 50 – 600 ÷ 10 = 25 – 60= -35

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Exercise 5 1. Choose the correct properties as discussed to complete the following:

a) 25 x 5 + 25 x 6 + 25 x 8

125 + 150 + 200 = 475

b) 1 00 ÷ 5 ÷ 20 ÷ 10

c) 1 019 – 63 + 214 – 1 000 + 50 + 35

d) 50 x 369

e) 2 ÷ 10 ÷ 4 x 20

2. Which of these are true and which are false?

If true, state which property you have used.

If false, write the correct method and/or answer.

a) 8 + 7 = 7 + 8

b) 10 x (7 +3 +4) = (10 x 7) + (10 x 3) + (10 x 4)

c) 5 + (10 – 3) = (5 + 10) – 3

d) (2 500 ÷ 20) ÷ 5 = 2 500 ÷ (20 ÷ 5)

3. Use any method to solve each of the following: 3.1. A grocer buys 480 trays of oranges. He has to share the oranges equally between

20 clients. How many trays will each client receive?

3.2. Mr Padaychee owns a clothing shop. He has 25 designer tops which he sells for

R69 each. Assuming Mr Padaychee sells all of the tops, how much money will he

receive?

3.3. Julia orders 32 tablecloths for the school. Each tablecloth costs R122. How much

does she have to pay?

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3.4. The Foundation Phase children entered a competition and won and amount of R34

545. The teachers decided to share the prize money between 350 children who

entered. How much money would each child get?

3.5. Angel uses her money to purchase the following items:

Apples R9.99

Potatoes R35,50

Bread R6,50

Milk R4,99

Chocolate R5,69

a) How much money does she have to pay?

b) If she pays with R100, how much change will she get?

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CONCEPT : FOLLOWING A SPECIFIC ORDER OF OPERATIONS

(BODMAS)

The order of operations for BODMAS is as follows:

B - Brackets

O - Of

D - Division

M - Multiplication

A - Addition

S - Subtraction

Study the equation below:

16 + 4 x 5 =

If you solve the problem from left to right, the answer is 100

(i.e. ( 16+ 4 ) x 5 = 100)

If you solve the problem according to BODMAS, the answer is 36.

(i.e. 16 + ( 4 x 5 ) = 36)

Both these answers seem correct.

HOW IS THIS POSSIBLE?

When we read words or numbers, we read from left to right. However, mathematicians

decided many years ago that a method should be used that orders operations in a specific

way. This is how BODMAS came about.

Remember, when doing BODMAS, that when multiplication and division are carried out,

you may start with whichever one of these operations comes first, in order from left to

right.

The same principle applies to addition and subtraction. You may start with whichever one

comes first, in order from left to right.

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Example:

235 + 80 x 5 ÷ 10 – 215

= 235 + 400 ÷ 10 – 215 (multiplication first)

= 235 + 40 – 215 (division)

= 275 – 215 (addition)

= 60 (subtraction)

(BODMAS)

NOW TRY THESE: 1. 30 X 12 ÷ 4 + ¾ 0f 20

2. ( 17 x 25 ) x ( 24 ÷ 2 ) – 10 x 10

3. 78 – 33 ÷ 3

4. ( 24 – 14 ) x 25 ÷ 5

5. ( 1 420 + 780 ) – (509 + 351 )

6. ( 710

of 500 ) x 48 ÷12 𝑥𝑥 2100 ÷25

Exercise 6 Solve the problems below, using BODMAS/BOMDAS.

No calculators are allowed. Show all working out.

a) 17 x ( 4 + 9 ) ÷ (23 – 10 )

b) 8 x 4 + 10 ÷ 2 x 3

c) 235 + 80 x 50 ÷ 10 – ( 215 + ¾ of 8 )

d) ( 65 – 15 ) x 3 + 21 ÷ 7

e) 200 ÷ 20 + 285 ÷ 95 – 8 + 4

f) 71 x 10 𝑥𝑥 10,44 𝑥𝑥 13

g) 790 + 1 000 ÷ 125 – 50 ÷ 10

h) 25 + 60 x 5 ÷ 10 – 50

i) 12 000 - 10 𝑥𝑥 10012𝑜𝑜𝑜𝑜 250

+ ½ of 100 + ( 200 ÷ 10 x 2 )

j) 2 + 93 𝑥𝑥 282 604 ÷12

+ 29 of 108

k) 50 + 35of 75 – 32

l) 470 + 692 x 10 ÷ 20 – 630

m) 38 of 4 000 + ½ of 250 – 10 ÷ 5

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Homework Task

n) 150 + ¾ of 24 + 27 ÷ 9 + 35 ÷7+409 𝑥𝑥 5

o) 164 + 36 x 5 – 64

p) 163 – 25 x ( 26 – 20 )

q) 49 – 15 x 2

r) 46 – 32 + 27 x ( 104 ÷ 8 )

s) 14412

+ 18 of 24 - 35 𝑥𝑥 2+10

108 ÷12+11

t) 8 967 - 240 ÷5 x 348 ÷4

+ 1012

of 48 + ( 9 x 6 )

VERTICAL METHOD ADDITION

H T U

3 8 4

5 2 7

+ 4 7 2

1 3 8 4

SUBTRACTION 7000

- 549

6451

Exercise 7 a) 95+362+285+274+1

b) 4 260+5 721+842+393

c) 33 333+55 555+77 777

d) 100 403+859 782

e) 357 901+ 129 042

f) 592 710 – 361 204

g) 240 040 - 67 952

h) 146 053 – 23 912 + 503 614

i) 13 420 + 118 066 – 120 786

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HORIZONTAL METHOD

Exercise 8 384 + 527 + 472 = (300+500+400) + (80+20+70) + (4+7+2)

= 1200+170+13

= 1 383

1) 456 +350 + 239 = (400+300+200)+(50+50+9)+(6+0+9)

= 900 + 109 +15

= 1000 + 9+15 = 1024

2) 648 + 352 + 371 =(600+300+300)+(40+50+70)+(8+2+1)

3) 872 + 287 + 451 =

4) 729 + 457 + 290 =

5) 452 + 628 + 303 =

6) 987–232–334 =

7) 798 – 362 – 234 =

7) 692 – 361 – 100 =

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MULTIPLICATION HORIZONTAL MULTIPLICATION 159 x 27 = (159x20) + (159x7)

= 20 x (100+50+9) + 7 x (100+50+9)

= (20x100) + (20x50) + (20x9) + (7x100) + (7x50) + (7x9)

= 2 000 + 1 000 + 180 + 700 + 350 + 63

= 3 180 + 1 113

= 4 293

235 x 34 =(235 x 30) + (235 x 4)

= 30 x (200+30+5) + 4 x (200+30+5)

= (30 x 200) + (30 x 30) + (30 x 5) + (4 x 200) + (4 x 30) + (4x5)

= 6000 + 900 + 150 + 800 + 120 + 20

= 7990

Exercise 9 b) 2 364 x 37 = (2364 x 30) + (2364 x 7)

= 30 x ( 2000 + 300+60+4) + 7 x ( 2000 + 300+60+4

= 60 000 + 9 000 + 1800 + 120 + 14 000 + 2100 + 420 + 28

= 74 000 + 12 900 + 568

= 87 468

c) 1 226 x 82 = (1226 x 80) + (1226 x 2)

d) 3 437 x 24 = (3437 x 20) + (3437 x 4)

e) 4 462 x 52 = (4 462 x 50) + (4 462 x 2)

f) 5 349 x 21 = (5 349 x 20) + (5 349 x 1)

g) 4 567 x 43 = (4 567 x 40) + (4 567 x 3)

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VERTICAL METHOD FOR MULTIPLICATION 159 x 27 = 6 8 9 x 3 4 =

1 5 9 6 8 9

X 2 7 x 3 4

1 1 1 3

3 1 8 0

4 2 9 3

1) 695 x 76 = 2) 478 x 38 =

3) 432 x 29 = 4) 659 x 37 =

Exercise 10 (Homework) 5) 246 x 37 =

6) 378 x 24 =

7) 425 x 42 =

8) 624 x 38 =

9) 732 x 26 =

10) 634 x 35 =

DIVISION - (LONG DIVISION)

576

9 |5 184

4 500

684

630

54

54

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EXERCISE 11

1) 350 ÷ 14

2) 1 512 ÷ 28

3) 5 168 ÷ 17

4) 2 106 ÷ 39

5) 9 936 ÷ 48

Homework DBE book 5a and 5b pg 12,13,14,15,

Using a calculator

Exercise 12

When using a calculator always remember to follow the BODMAS order of operations.

Try this: 10 + 3 x 2 = 305 x 56 – 14 082 =

1) 389 + 573 + 289 =

2) 689 – 175 – 243 =

3) 429 x 37 =

4) 504 ÷ 56 =

5) 1 034 x 26 =

6) 8 322 ÷ 73 =

7) 5 893 x 45 =

8) 13 132 ÷ 134 =

9) 5 077 + 3 240 + 652 – 8 003 =

10) 7 128 ÷ 33 – 216 =

Homework - Using a calculator 1) 367 + 35 x 89 =

2) 3 005 – 21 x 42 =

3) 2 876 – 12 948 ÷ 83 =

4) 27 590 - 35÷ 328 ÷ 23 =

5) 3 052 – (415 + 78) =

6) 8 427 + (34 x 73) =

7) 14 775 – (39 x 258) =

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8) 3 083 + (28 635 ÷ 23) =

9) 384 x 75 + 179 x 202 =

RATIO AND RATE.

A. DEFINITIONS

1. RATIO • A ratio is used to compare the sizes of two or more quantities that use the same unit of

measurement.

• A ratio of 5 : 6 means that for every 5 of the first quantity, there are 6 of the second quantity.

• Ratio can also be written as a fraction. In the ratio 5 : 6, the first quantity would be written as 511

.

The second quantity would be written as 611

.

• Ratios can be simplified.

e.g. 10 : 12 can be simplified to 5 : 6.

2. RATE

• A rate compares 2 or more quantities that use different units of measurement, e.g. km/l.;

R/kg.

EXERCISE 13

Calculating Ratio

1. Write ratios for the following given amounts and then simplify each ratio where possible:

1.1. There are 4 girls and 6 boys in a nursery school group.

1.2. There are 5 roses in a bunch of 20 flowers.

1.3. There are 20 kg of vegetables for every 15 kg of fruit.

1.4. There are 32 teachers and 1 024 children in a school.

1.5. 2 parts of cold drink can be mixed with 10 parts of water.

2. Simplify these ratios

2.1) 15 : 35 2.2) 20 : 30 2.3) 3:6:12

2.4) 24 : 60 : 84 2.5) 10 cm : 2 m 2.6) 40 m : 2km

Ratio Activity DBE book 23 pg 62 & 63

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EXERCISE 14

Calculating Ratio

1.1. A chef bakes a dozen cookies in 20 minutes. How many cookies does he bake in 3 hours?

1.2. A seamstress sews 16 tops in 1 hour. How many tops does she sew in: a) half an hour? b) 9 hours? 1.3. 500 pairs of shoes are sold for a total R2 500. How much does 1

pair of shoes cost? 1.4. A truck driver takes 4, 5 hour to travel 400 km. What is the average speed per

hour? 1.5. 14 Sharpeners cost a total of R35. What will 6 sharpeners cost? 1.6. 4 passengers pay R380 as travel fare. What is the fee per passenger? 1.7. 6 kg of biltong costs R462,00. How much would you pay for 2,5 kg of biltong? 1.8. A car travels 360 km in 4 hours. How far does it travel in 6 hours? 1.9. Kyra must take 15 ml of medicine every 6 hours. How much medicine must Kyra

take daily? 1.10. A butcher sells 15 kg of mince for R900,00. How much does the mince cost per

kilogram? 2. Simplify these ratios :

a) 15 :35 = b) 20:30 =

c) 3: 6 :12 = d) 24: 60 : 84 =

e) 10cm : 2m = f) 40m :2km =

3. An ID photograph measures 2 cm by 3 cm. Complete the table below, using the same

ratio:

4. Simplify the following ratios:

a) 12 : 40

b) 38 : 90

c) 25 : 1 000

d) 50 c : R10,00

e) 40g : 1kg

f) 8 l: 250 ml

g) 9 hours : 150 minutes

h) 100 m : 5 km

i) R7,50 : 25 c

j) 8 m : 25cm

Type of photo Width Length Ratio

Digital 12 2 : 3

Wedding 24 2 : 3

Portrait 96

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Sharing a “whole” in a given ratio.

Share R 2 250 in the ratio 3:2:1

• This means 3:2:1 that 3+2+1= 6 parts of the whole 2 250.

• In fraction form, this means 36 of 2 250

= 3x2 250

= 6 750 ÷ 6

= R 1 125

• 26 of 2 250

= 2x2 250

= 4 500 ÷ 6

= R 750

• 1 6 of 2 250

= 2 250 ÷ 6

= R 375

EXERCISE 15

1) Divide R 200 between you and your best friend in the ratio 3:2

2) Divide R 240 in the ratio 3:4:5

3) Share 28 sweets between Joe and Amy in the ratio 3:1

4) Share an inheritance of R 50 000 between five children in the following ratio 7:9:3:2:4

Homework DBE book 24 pg 64 & 65

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CONCEPT :

CALCULATING PERCENTAGE INCREASE AND DECREASE

• When increasing or decreasing a number by a given percentage, write the

percentage out of 100 and multiply it by the given number.

Example: Increase R 1 500 by 25%

= 10025 x

11500 *Simplify / Cancel if possible

= R 375

Now add this amount to the original value:

i.e. R1500+R375

= R 1 875

• If decreasing, you would subtract this amount from the original value

Example: Decrease R 3 000 by 45%

= 10045 x

13000

= R 1 350

Decreased amount: R3000–R1350

= R1650

EXERCISE 16

Solve the problems below:

1. The building cost of houses has increased by 20%. If a house previously cost

R 26 000, what will the new building cost be?

2. Civil servants are told they will get a salary increase of 4%. How much will they earn if

their current salaries are R 4 500?

3. The value of a home entertainment system decreased by 15% after 1 year. If the price

is R 17 800, what will its value be after a year?

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4. The Grade 7 learners from last year made a profit of R 18 560 on Entrepreneurs’ Day.

This year’s Grade 7 learners made 12% more profit than the previous group. How

much profit was made?

5. Thandiwe bought a book that cost R 52. If she was given a cash discount of 8%, how

much will she pay for the book?

6. Increase R 4 575 by 25%.

7. Decrease R 7 500 by 40%.

8. I bought a CD for R 120. I sell it for 15% less than what I paid for it. How much money

will I receive?

CONCEPT : CALCULATING VAT, INTEREST AND DISCOUNT

• When calculating VAT, 14% of the article’s value must be added to the cost of the

article (i.e. increase by 14%).

• When calculating discount, we decrease the value of the article by the given

percentage.

• Interest is added to the value of an article, normally when credit is given and the

buyer cannot pay the full amount.

Example 1:

Calculate the VAT on the article costing R 500

= 10014 x

1500

= 70

=R500+R70

= R 570

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Example 2:

Calculate what you will pay for an item costing R 250 if a discount of 15% is given.

10015 x

1250

= 275

= R 37,50

= R250–R37,50

= R 212,50

CONCEPT : PROFIT AND LOSS

A. WHAT IS PROFIT AND LOSS?

• PROFIT is when an item is sold for more than what it

originally cost.

• LOSS is when an item is sold for less than what it

originally cost.

B. HOW DO WE CALCULATE PROFIT OR LOSS?

Step 1: Calculate the profit or loss by working out the

difference between the cost price and the selling price

of an article.

Step 2: Write this amount as a fraction over the cost price and

multiply it by 1

100

Step 3: The answer will tell you what percentage profit loss

was made.

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Example 1: I buy an article for R 300 and sell it for R 500. What

percentage profit did I make?

Step 1: R 500 – R 300 = R 200

Step 2: 300200 x

1100

= 3

200

Step 3: = 66,6% profit

Example 2: I buy an article for R 75 and sell it for R 40. What

percentage loss did I make?

Step 1: R 75 – R 45

= R 35

Step 2: 7535 x

1100

Step 3: = 3

140

= 46,6% loss

CALCULATING VAT, INTEREST, DISCOUNT and profit and loss

EXERCISE 17

Work out the percentage profit or loss for each of the problems below:

1. A music shop buys a CD player for R 450 and sells it for R 300.

2. A student buys a motorbike for R 20 500 and sells it for R 17 700.

3. A seamstress buys material for R 87,50 and sells the completed garment for R 165.

4. A wooden showcase is bought for R 1 500 and sold for R 3 750.

5. A fridge is bought at a cost of R 2 999 and sold years after for R 1000.

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6. I buy a leather jacket for R 1 899 and sell if for R 2 200.

7. A painting is purchased from the artist at a cost of R 595. An art dealer then sells the

painting to a customer for R 2 135,50.

8. Shoes that originally cost R 185 are now marked down to R 99.

9. A computer programmer tries to install a programme that costs him R 1 350, but finds that

it is faulty and sells it for R 750.

10. I make up a chocolate hamper that cost me R 45,60 and sell it for R 85.