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PRY 6 MATHEMATICS THIRD TERM

WEEK 2

OPEN SENTENCE

(a). If 40 notes are to be shared Among 5 pupils, how many books Will be given to a pupil?

5   pupils 40 notes

1   pupils (40 + 5) notes

= 8 notes

(b). Find the letters e.g.

 

2y + 6 = 30

2y = 30 –6

2y = 24 divide both sides By 2

2Y/2=24/2

= y = 12

Closed and open sentences

Study the following mathematical statements:

13 + 6 = 19 23 + 12 = 35

42 − 20 = 22 63 − 49 = 14

7 × 5 = 35 11 × 12 = 132

40 ÷ 5 = 8 120 ÷ 10 = 12

The mathematical statements above are called closed number sentences.

Closed number sentences can either be true or false.

Examples

15 + 7 = 22 (True mathematical statement) 18 + 3 = 19 (False mathematical statement)

3 × 6 = 12 (False mathematical statement) 42 ÷ 6 = 7 (True mathematical statement)

Study each of the following mathematical statements:

[]+ 9 = 13   11 +[] = 25   [] − 4 = 11 20 − = 7

[]× 5 = 15     4 ×[] = 24    [] ÷ 6 = 5 48 ÷ = 12

In each of the statements above, there is a missing number called unknown represented by

.[] They are called open sentences.

An open sentence is a mathematical statement that involves equality signs and a missing

quantity represented by[] that the four arithmetic operations of addition, subtraction,

multiplication and division can be applied to solve.

Open sentences can either be true or false depending on the value [].

Exercise

  1. Write True (T) or False (F) for each of the following closed number sentences.
  2. 15 + 16 = 31 2. 54 + 4 = 68 3. 18 + 10 = 38   4. 51 + 47 = 98
  3. 29 + 60 = 82 6. 42 + 54 = 84 7. 55 − 23 = 33  8. 54 − 11 = 43
  4. 64 − 43 = 21 10. 98 − 45 = 53
  5. Write True (T) or False (F) for each of the following open sentences if is replaced by 4.

1.[] + 2 = 9      2[]. + 3 = 7     3.[] + 7 = 12     4.[] − 3 = 1

  1. 12 –[] = 7 6. 8 –[] = 4 7. 4 × []= 16     8.[] × 2 = 10

9.[] ÷ 2 = 2

Operation of addition and subtraction involving open sentences (Revision)

Examples

Here the number represented by in each of the following has been found.

1.[] + 14 = 36   2. 12 +[] = 8    3[]. − 4 = 30    4. 15 –[] = 9

Solution

1.[] + 14 = 36 can be interpreted as “what can be added to 14 to get 36?”

[]+ 14 = 20 + 16

[]+ 14 = 20 + 2 + 14

[]+ 14 = 22 + 14

[]= 22

Check:

22 + 14 = 36

Short method

If[] + 14 = 36

then []= 36 − 14

= 22

= 22

Check:

22 + 14 = 36

  1. 12 +[] = 20 + 10

12 +[] = 12 + 8 + 10

12 + []= 12 + 18

= 18

Check:

12 + 18 = 30

Short method

If 12 +[] = 30

Then[] = 30 – 12

= 18

= 18

Check:

12 + 18 = 30

Note: Since the problem is addition, the number is subtracted from each other to find .

3 [] . − 4 = 8 can be interpreted as “what number minus 4 gives 8?”

[]− 4 = 12 – 4

[]= 12

Check:

12 − 4 = 8

Short method

If []− 4 = 8

Then[] = 8 + 4

= 12

Check:

12 − 4 = 8

Note: The numbers 8 and 4 are added to get the number represented by[] .

  1. 15 –[] = 9 can be interpreted as ‘when a number is subtracted from 15, the answer is 9’

15 –[] = 9

15 –[] = 15 – 6 [15 = 9 + 6]

[]= 6

Check:

15 − 6 = 9

Short method

If 15 –[] = 9

Then[] = 15 – 9

= 6

Check:

15 − 6 = 9

Note: 9 is subtracted from 15 to get the number represented by []

Exercise

  1. Find the number represented by in each of the following.
  2. 9 +[] = 16 2[]. + 25 = 34 3[]. + 3 = 14
  3. 8 = 5 +[] 5[] + 17 = 25 6. 7 +[] = 13
  4. Find the number represented by in each of the following.

1.[] − 16 = 13      2[]. − 7 = 23     3. 19 –[] = 11

  1. 77 =[] – 39 5. 17 =[] – 59 6[]. − 17 = 39

Operation of multiplication and division involving open sentences (Revision)

Examples

Find the number represented by in each of the following:

  1. 7 ×[] = 56 2.[] × 4 = 48 3. 60 ÷[] = 12   4[]. ÷ 8 = 9

Solution

  1. 7 × = 56 can be interpreted as “7 multiplied by a certain number equals 56”

7 ×[] = 7 × 8

= 8

Check:

7 × 8 = 56

Short method

If 7 × = 56

then =

56/7

=8 × 7/7 = 8

Check:

7 × 8 = 56

  1. × 4 = 12 × 4

= 12

Check:

12 × 4 = 48

Short method

If × 4 = 48

then =

48/4

= 12 × 4/4 = 12

Check:

12 × 4 = 48

  1. 60 ÷ = 12 can be interpreted as

‘what number divides 60 gives 12?’

60 ÷ = 12

60 = 5 × 12

60 ÷ 5 = 12, 60 ÷ 12 = 5

60 ÷ = 60 ÷ 5

= 5

Check:

60 ÷ 5 = 12

  1. ÷ 8 = 9 can be interpreted as ‘when

a number is divided by 8, the answer is 9’

÷ 8 = 9

÷ 8 = 72 ÷ 8

= 72

9 × 8 = 72

72 ÷ 8 = 9

72 ÷ 9 = 8 Check: 72 ÷ 8 = 9

Exercise

Find the number represented by in each of the following.

  1. 6 × = 48 2. × 8 = 96 3. × 5 = 45 4. 6 × = 60
  2. 4 × = 36 6. × 4 = 28 7. × 11 = 33 8. 12 × = 84
  3. ÷ 5 = 7 10. 14

of = 16 11. 12

of = 18 12. 3 × = 18

  1. ÷ 8 = 32 14. 1

10 of = 9 15. 680 ÷ = 34 16. 13

of = 12

  1. 448 ÷ = 56 18. 1

10 of = 42

Use of letters to find the unknown

Activity

Study the following mathematical statements.

+ 5 = 11, a + 5 = 11 6 + = 15, 6 + y = 11 − 3 = 2, x − 3 = 2

× 2 = 12, 2 × m = 12 32 ÷ = 8, 32 ÷ n = 8

By comparing each statement, you will discover that the box is replaced with a letter of

the alphabet. That is;

+ 5 = 11 is the same as a + 5 − 3 = 2 is the same as x − 3 = 2 and so on.

Mathematical statements containing simple letters and numbers are called simple equations.

When the value of the letter is solved, the equation is solved.

Examples

  1. x + 5 = 12 2. y − 12 = 3 3. 2m = 14 4. a5

= 6

Hint: Write a sentence to show the meaning of each equation.

Solution

  1. x + 5 = 12 can be interpreted as “If a number is added to 5 we get 12”
  2. 2m = 14 (2 m means 2 × m) can be interpreted as ‘what number multiplied by 2 gives 14?’

2 × m = 2 × 7

m = 7

Check:

2m = 2 × m = 2 × 7 = 14

Short method

If 2m = 14

then m = 14

2

= 7

Check: 2m = 2 × m = 2 × 7 = 14

a5

= 6 5 × 6 = 30

a5

= 30

5 30 ÷ 5 = 6, 30 ÷ 6 = 5 a = 30

Check: a5

= 30

5 = 6 5 × 6 = 30

Short method

If a5

= 6

then a = 5 × 6 = 30

  1. y − 12 = 3 can be interpreted as “If 12 is subtracted from a number, the answer is 3”

x + 5 = 7 + 5

x = 7

Check:

7 + 5 = 12

Short method

If x + 5 = 12

then x = 12 − 5

= 7

Check:

x + 5 = 7 + 5 = 12

y − 12 = 3

y − 12 = 15 − 12

y = 15

Check:

15 − 12 = 3

Short method

If y − 12 = 3

then y = 3 + 12

= 15

Check:

y – 12 = 15 − 12 = 3

Check: a5

= 30

5 = 6

  1. a5

= 6 can be interpreted as ‘when a number is divided by 5 we get 6’

Exercise

Solve the following equations.

  1. m + 5 = 8 2. p + 6 = 13 3. d + 8 = 17 4. c + 2 = 12
  2. e + 8 = 18 6. 5 + x = 9 7. 1 + q = 25 8. 12 + t = 30
  3. m − 6 = 13 10. p − 5 = 15 11. q − 7 = 21 12. k − 12 = 35

Examples

  1. Think of a number, add 7 to it, and the result is 21. Study how the number is found.

Word problems

Solution

The number I think of + 7 = 21

Let m stand for the unknown number then,

m + 7 = 21

m + 7 = 10 + 10 + 1

m + 7 = 11 + 3 + 7

m + 7 = 14 + 7 m = 14

Short method

m + 7 = 21

m = 21 − 7

= 14

Check:

m + 7 = 14 + 7

= 21

  1. If 43 is subtracted from a number, we get 38. Study how the number is found.

Solution

Unknown number − 43 = 38

Let x stand for the unknown number, then

x − 43 = 38

x − 43 = 81 − 43

x = 81

Short method

x – 43 = 38

x = 38 + 43 = 81

Check:

x – 43 = 8 1

− 4 3

3 8

  1. I think of a number, multiply it by 3 and the result is 36. Study how the number is found.

Solution

Unknown number × 3 = 36

Let y be the unknown number, then

y × 3 = 36

y × 3 = 12 × 3

y = 12

  1. When a number is divided by 7 we get 9. Study how the number is found.

Solution

Unknown number ÷ 7 = 9

Let q be the unknown number

q ÷ 7 = 9

q ÷ 7 = 63 ÷ 7 q = 63

7 × 9 = 63

63 ÷ 7 = 9

63 ÷ 9 = 7

Check: q ÷ 7 = 63 ÷ 7 = 9

Exercise

  1. When 79 is added to a number, we get 124. Find the number.
  2. When 71 is added to a number, we get 214. Find the number.
  3. When I subtract 19 12

from a certain number, the result is 9 12

. What is the number?

  1. When 31 kg of meat is removed from the part of the cow, there is 25 kg left. What is the

weight of the cow?

  1. A poultry farmer took four crates of eggs to the market. He had 45 eggs left after

market hour. How many eggs were sold?

  1. When 564 is added to a certain number, the result is 801. Find the number.
  2. 6 times an unknown number gives 72. Find the number.
  3. When a number is multiplied by 12, we get 108. Find the number.
  4. I think of a number, divide it by 8 and get 32. Find the number.
  5. A certain number of oranges was shared equally among 6 children. Each child

received 14 oranges. How many oranges were shared?

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