Operations with Integers

Page 25: Sums & Differences

Question

Find pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13

How to do it:

Let the two numbers be x and y.

x + y = Sum

x - y = Difference

Add both equations:

2x = Sum + Difference

x = (Sum + Difference) / 2

Then find y: y = Sum - x

(a)

Sum = 27, Diff = 9

x = (27 + 9) / 2 = 36 / 2 = 18

y = 27 - 18 = 9

Answer (a): 18 and 9

(b)

Sum = 4, Diff = 12

x = (4 + 12) / 2 = 16 / 2 = 8

y = 4 - 8 = -4

Answer (b): 8 and -4

(c)

Sum = 0, Diff = 10

x = (0 + 10) / 2 = 10 / 2 = 5

y = 0 - 5 = -5

Answer (c): 5 and -5

(d)

Sum = 0, Diff = -10

x = (0 + (-10)) / 2 = -10 / 2 = -5

y = 0 - (-5) = 5

Answer (d): -5 and 5

(e)

Sum = -7, Diff = -1

x = (-7 + (-1)) / 2 = -8 / 2 = -4

y = -7 - (-4) = -3

Answer (e): -4 and -3

(f)

Sum = -7, Diff = -13

x = (-7 + (-13)) / 2 = -20 / 2 = -10

y = -7 - (-10) = 3

Answer (f): -10 and 3

Page 31: Token Interpretations

Question 1(a)

Using the token interpretation, find the value of: 3 × (-2)

Multiplier is positive (3).

So we PUT IN tokens into an empty bag.

Put in 2 negative tokens, three times.

Count the bag: 6 negative tokens.

Answer: -6

Question 1(b)

Using the token interpretation, find the value of: (-5) × (-2)

Multiplier is negative (-5).

So we TAKE OUT tokens from an empty bag.

We need to take out 2 negative tokens, five times.

Bag is empty, so first add 10 zero-pairs.

Then cross out the 10 negative tokens.

What remains: 10 positive tokens.

Answer: 10

Question 1(c)

Using the token interpretation, find the value of: (-4) × (-1)

Multiplier is negative (-4).

So TAKE OUT 1 negative token, four times.

Add 4 zero-pairs to the empty bag.

Cross out the 4 negatives.

What remains: 4 positive tokens.

Answer: 4

Question 1(d)

Using the token interpretation, find the value of: (-7) × 3

Multiplier is negative (-7).

So TAKE OUT 3 positive tokens, seven times.

Add 21 zero-pairs to the empty bag.

Cross out all 21 positives.

... (repeated 21 times total) ...

What remains: 21 negative tokens.

Answer: -21

Question 2 & 3

If 123 × 456 = 56088, find without calculating:
(a) (-123) × 456
(b) (-123) × (-456)
(c) (123) × (-456)
Also, try to frame a simple rule to multiply two integers.

(a)

One number is negative.

So product is negative.

(-123) × 456 = -56088

(b)

Both numbers are negative.

So product is positive.

(-123) × (-456) = 56088

(c)

One number is negative.

So product is negative.

123 × (-456) = -56088

Simple Rule:

Multiply the absolute values.

Same signs → product is positive.

Different signs → product is negative.

Page 39: Division & Word Problems

Question 1

Find the values of: (a) 14 × (-15), (b) -16 × (-5), (c) 36 ÷ (-18), (d) (-46) ÷ (-23)

(a) Different signs → negative

14 × (-15) = -210

(b) Same signs → positive

-16 × (-5) = 80

36 ÷ (-18) = -2

(d) Same signs → positive

(-46) ÷ (-23) = 2

Question 2

A freezing process requires that the room temperature be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Initial temperature = 32°C

Temperature is lowering → rate is negative

Rate = -5°C per hour

Total change in 10 hours = 10 × (-5) = -50

Final temperature = 32 + (-50) = 32 - 50 = -18

Answer: -18°C

Question 3(a)

A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold. The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?

Profit on white = +8 per bag

Loss on grey = -5 per bag

Total white profit = 3000 × 8 = +24000

Total grey loss = 5000 × (-5) = -25000

Net = 24000 + (-25000) = -1000

Answer: Loss of ₹1000

Question 3(b)

If the number of bags of grey cement sold is 6,400 bags, what is the number of bags of white cement the company must sell to have neither profit nor loss?

Grey bags = 6400

Total grey loss = 6400 × (-5) = -32000

To have no profit no loss, white profit must cancel this loss.

So white profit needed = +32000

Let number of white bags = x

x × 8 = 32000

x = 32000 / 8 = 4000

Answer: 4000 white cement bags

Question 4

Replace the blank with an integer to make a true statement:
(a) (-3) × __ = 27
(b) 5 × __ = (-35)
(c) __ × (-8) = (-56)
(d) __ × (-12) = 132
(e) __ ÷ (-8) = 7
(f) __ ÷ 12 = -11

(a) (-3) × ? = 27

Different signs needed for positive product? Wait, both negative.

(-3) × (-9) = 27

(b) 5 × ? = -35

Product negative → blank must be negative.

5 × (-7) = -35

Product negative → blank positive.

7 × (-8) = -56

(d) ? × (-12) = 132

Product positive → blank negative.

(-11) × (-12) = 132

(e) ? ÷ (-8) = 7

Quotient positive → blank negative.

(-56) ÷ (-8) = 7

(f) ? ÷ 12 = -11

Quotient negative → blank negative.

(-132) ÷ 12 = -11

Pages 42-44: Advanced Expressions

Question 1

Find the values of the following expressions:
(a) (-5) × (18 + (-3))
(b) (-7) × 4 × (-1)
(c) (-2) × (-1) × (-5) × (-3)

(a) First solve inside brackets:

18 + (-3) = 15

Then (-5) × 15 = -75

(b) First (-7) × 4 = -28

Then (-28) × (-1) = 28

(-2) × (-1) = 2

(-5) × (-3) = 15

2 × 15 = 30

Question 2 & 3

Find the values of: (a) (-27) ÷ 9 (b) 84 ÷ (-4) (c) (-56) ÷ (-2)
Find the integer whose product with (-1) is: 27, -31, -1, 1, 0

2(a) Different signs → negative

(-27) ÷ 9 = -3

2(b) Different signs → negative

84 ÷ (-4) = -21

2(c) Same signs → positive

(-56) ÷ (-2) = 28

3. Multiplying by -1 just changes the sign:

27 × (-1) = -27

-31 × (-1) = 31

-1 × (-1) = 1

1 × (-1) = -1

0 × (-1) = 0

Question 4

If 47 - 56 + 14 - 8 + 2 - 8 + 5 = -4, then find the value of -47 + 56 - 14 + 8 - 2 + 8 - 5 without calculating the full expression.

Look carefully at the second expression.

Every single sign is the opposite of the first one.

That means the whole second expression is just (-1) times the first expression.

First expression = -4

So second = (-4) × (-1) = 4

Answer: 4

Question 5

Collatz Conjecture modification: If even, take half of it; if odd, multiply it by -3 and add 1. Try this with starting numbers: (-21) and (-6).

Start with -21 (odd):

(-21 × -3) + 1 = 63 + 1 = 64

64 is even → 32

32 → 16 → 8 → 4 → 2 → 1

Start with -6 (even):

-6 ÷ 2 = -3

-3 is odd → (-3 × -3) + 1 = 9 + 1 = 10

10 → 5

5 is odd → (5 × -3) + 1 = -15 + 1 = -14

-14 → -7

-7 is odd → (-7 × -3) + 1 = 21 + 1 = 22

22 → 11

11 is odd → (11 × -3) + 1 = -33 + 1 = -32

-32 → -16 → -8 → -4 → -2 → -1

-1 is odd → (-1 × -3) + 1 = 3 + 1 = 4

4 → 2 → 1

Question 6

In a test, (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer.
(a) Anita scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? Total questions?
(b) Anil scored (-10) marks even though he had 5 correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?

(a) Anita

15 correct → marks from correct = 15 × 4 = 60

Let wrong answers = x

Marks from wrong = -2x

Total score: 60 + (-2x) = 40

-2x = 40 - 60 = -20

x = 10

Total questions = 15 + 10 = 25

Answer (a): 10 incorrect, 25 total

(b) Anil

5 correct → marks from correct = 5 × 4 = 20

Let wrong answers = y

20 + (-2y) = -10

-2y = -10 - 20 = -30

y = 15

Questions attempted = 5 + 15 = 20

Test has 25 questions, so left unanswered = 25 - 20 = 5

Answer (b): 15 incorrect, left 5 unanswered

Question 7 & 8 & 9

7. Find the operations done by the machine: Input: 4, 8, -3 → Output 28. Find output for: -16, -6, -9.
8. If the temperature is currently at 8°C and drops 5°C each hour, write an expression for the temperature after 4 hours.
9. Find 3 consecutive numbers with a product of (a) -6, (b) 120

Check: 4 - (8 × -3) = 4 - (-24) = 4 + 24 = 28

So rule is: first number - (second × third)

Now for -16, -6, -9:

-16 - (-6 × -9) = -16 - 54 = -70

Starts at 8°C

Drops 5°C each hour → -5 per hour

After 4 hours: 8 + (4 × -5) = 8 - 20 = -12

9(a)

Three consecutive: -3, -2, -1

(-3) × (-2) × (-1) = -6

9(b)

Three consecutive: 4, 5, 6

4 × 5 × 6 = 120

Question 10

An alien society uses a peculiar currency called 'pibs' with just two denominations of coins - a +13 pibs coin and a -9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs? Try to get totals: (a)+20 (b)+40 (c)-50 (d)+8 (e)+10 (f)-2 (g)+1 (h) 1568 pibs

We need: 13x - 9y = target

(x = number of +13 coins, y = number of -9 coins)

For +85:

13(10) - 9(5) = 130 - 45 = 85 → Yes possible

(a) 20: 13(5) - 9(5) = 65 - 45 = 20

(b) 40: 13(10) - 9(10) = 130 - 90 = 40

(c) -50: 13(1) - 9(7) = 13 - 63 = -50

(d) 8: 13(2) - 9(2) = 26 - 18 = 8

(e) 10: 13(7) - 9(9) = 91 - 81 = 10

(f) -2: 13(4) - 9(6) = 52 - 54 = -2

(g) 1: 13(7) - 9(10) = 91 - 90 = 1

(h) 1568: Yes, because gcd of 13 and 9 is 1, so any integer is possible.

Question 11 & 12

11. Find the values of: (a) (32 × (-18)) ÷ ((-36)) (b) (32) ÷ ((-36) × (-18)) (c) (25 × (-12)) ÷ ((45) × (-27)) (d) (280 × (-7)) ÷ ((-8) × (-35))
12. Arrange the expressions below in increasing order: (a) (-348)+(-1064) (b) (-348)-(-1064) (c) 348-(-1064) (d) (-348) × (-1064) (e) 348 × (-1064) (f) 348 × 964

11(a)

32 × (-18) = -576

-576 ÷ (-36) = 16

11(b)

(-36) × (-18) = 648

32 ÷ 648 = 4/81

11(c)

25 × (-12) = -300

45 × (-27) = -1215

-300 ÷ -1215 = 20/81

11(d)

280 × (-7) = -1960

(-8) × (-35) = 280

-1960 ÷ 280 = -7

12. Values:

a = -348 + (-1064) = -1412

b = -348 - (-1064) = -348 + 1064 = 716

c = 348 - (-1064) = 348 + 1064 = 1412

d = (-348) × (-1064) = 370272

e = 348 × (-1064) = -370272

f = 348 × 964 = 335472

Order (smallest to largest): e, a, b, c, f, d

Question 13 & 14

13. Given that (-548) × 972 = -532656, write the values of: (a) (-547) × 972 (b) (-548) × 971 (c) (-547) × 971
14. Given that 207 × (-33 + 7) = -5382, write the value of -207 × (33 - 7)

13(a)

(-547) × 972 = (-548 + 1) × 972

= (-548) × 972 + 1 × 972

= -532656 + 972 = -531684

13(b)

(-548) × 971 = (-548) × (972 - 1)

= (-548) × 972 - (-548) × 1

= -532656 + 548 = -532108

13(c)

(-547) × 971 = (-547) × (972 - 1)

= (-547) × 972 - (-547)

= -531684 + 547 = -531137

14.

Notice: -207 × (33 - 7) = -207 × 26

But 207 × (-33 + 7) = 207 × (-26) = -5382

So -207 × 26 is the same as 207 × (-26)

Therefore answer is also -5382

Question 15 & 16

15. Use the numbers 3, -2, 5, 6 exactly once and the operations '+', '-', and '×' exactly once and brackets as necessary to write an expression such that (a) the result is the maximum possible (b) the result is the minimum possible.
16. Fill in the blanks in at least 5 different ways with integers: (a) [ ] + [ ] × [ ] = -36 (b) ([ ] - [ ]) × [ ] = 12 (c) ([ ] - ([ ] - [ ])) = -1

15(a) Maximum:

Make the positive parts as large as possible.

(6 - (-2)) × (5 + 3) = 8 × 8 = 64

15(b) Minimum:

Make a large negative number.

(-2 - 6) × (5 + 3) = (-8) × 8 = -64

16. Some possible ways:

(a) 0 + 6 × (-6) = -36

(b) (5 - 2) × 4 = 12

(c) 1 - (4 - 2) = 1 - 2 = -1