Operations with Integers | FIO

Question 2

Using the token interpretation, find the values of:

(a) $3 \times (-2)$

(b) $(-5) \times (-2)$

(d) $(-7) \times 3$

Solution

Understand the Question

In the token model:
- A positive token (yellow) represents $+1$ and a negative token (red) represents $-1$ .
- A zero pair consists of one positive token and one negative token with a net value of $0$ .
In a multiplication $a \times b$ $a \times b$ :
- The first number $a$ indicates whether to add (if positive) or remove (if negative) groups of tokens.
- The second number $b$ indicates the type and number of tokens in each group.
- When removing tokens from an empty board, start by adding enough zero pairs so the required tokens can be taken away.

(a) Using the token interpretation, find the value of $3 \times (-2)$

Step 1 · Add 2 Negative Tokens 3 Times

Start with an empty board and add $2$ negative tokens, $3$ times. Diagram 1

\begin{aligned} 3 \times (-2) &= \text{add 2 negative tokens, 3 times} \\ &= \text{6 negative tokens} \\ &= -6 \end{aligned}

Answer

(a) $-6$

(b) Using the token interpretation, find the value of $(-5) \times (-2)$

Step 1 · Remove 2 Negative Tokens 5 Times

To remove $2$ negative tokens $5$ times (total of $10$ negative tokens), add $10$ zero pairs to the board and remove the $10$ negative tokens. Diagram 2

\begin{aligned} (-5) \times (-2) &= \text{remove 2 negative tokens, 5 times} \\ &= \text{add 10 zero pairs, then remove 10 negative tokens} \\ &= \text{10 positive tokens} \\ &= 10 \end{aligned}

Answer

(b) $10$

(c) Using the token interpretation, find the value of $(-4) \times (-1)$

Step 1 · Remove 1 Negative Token 4 Times

To remove $1$ negative token $4$ times (total of $4$ negative tokens), add $4$ zero pairs to the board and remove the $4$ negative tokens. Diagram 3

\begin{aligned} (-4) \times (-1) &= \text{remove 1 negative token, 4 times} \\ &= \text{add 4 zero pairs, then remove 4 negative tokens} \\ &= \text{4 positive tokens} \\ &= 4 \end{aligned}

Answer

(c) $4$

(d) Using the token interpretation, find the value of $(-7) \times 3$

Step 1 · Remove 3 Positive Tokens 7 Times

To remove $3$ positive tokens $7$ times (total of $21$ positive tokens), add $21$ zero pairs to the board and remove the $21$ positive tokens. Diagram 4

\begin{aligned} (-7) \times 3 &= \text{remove 3 positive tokens, 7 times} \\ &= \text{add 21 zero pairs, then remove 21 positive tokens} \\ &= \text{21 negative tokens} \\ &= -21 \end{aligned}

Answer

(d) $-21$

Common Mistakes

Forgetting Zero Pairs: Trying to remove tokens directly from an empty board without first introducing the necessary number of zero pairs.
Sign of the Product: Confusing removal with addition—removing negative tokens leaves positive tokens behind, which is why $(\text{negative}) \times (\text{negative}) = \text{positive}$ .

More questions in FIO

Let us try to find a few more pairs of numbers from their sums and differences:

(a) Sum $= 27$ , Difference $= 9$

(b) Sum $= 4$ , Difference $= 12$

(d) Sum $= 0$ , Difference $= -10$

(e) Sum $= -7$ , Difference $= -1$

(f) Sum $= -7$ , Difference $= -13$

Using the token interpretation, find the values of:

(a) $3 \times (-2)$

(b) $(-5) \times (-2)$

(d) $(-7) \times 3$

If $123 \times 456 = 56088$ , without calculating, find the value of:

(a) $(-123) \times 456$

(b) $(-123) \times (-456)$

Try to frame a simple rule to multiply two integers.

Consider the numbers represented by the following tokens in the diagram:

Find the following products.

(a) $4 \times (-3)$

(b) $(-6) \times (-3)$

(d) $(-8) \times 4$

(e) $(-9) \times 10$

(f) $10 \times (-17)$

Find the values of:

(a) $14 \times (-15)$

(b) $-16 \times (-5)$

(d) $(-46) \div (-23)$

A freezing process requires that the room temperature be lowered from $32^\circ\text{C}$ at the rate of $5^\circ\text{C}$ every hour. What will be the room temperature 10 hours after the process begins?

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. [Represent the profit/loss as integers.]

(a) 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? (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?

Replace the blank with an integer to make a true statement.

(a) $(-3) \times \underline{\quad\quad} = 27$

(b) $5 \times \underline{\quad\quad} = (-35)$

(d) $\underline{\quad\quad} \times (-12) = 132$

(e) $\underline{\quad\quad} \div (-8) = 7$

(f) $\underline{\quad\quad} \div 12 = -11$

Q10

Find the values of the following expressions:

(a) $(-5) \times (18 + (-3))$

(b) $(-7) \times 4 \times (-1)$

Q11

Find the values of the following expressions:

(a) $(-27) \div 9$

(b) $84 \div (-4)$

Q12

Find the integer whose product with $(-1)$ is:

(a) $27$

(b) $-31$

(d) $1$

(e) $0$

Q13

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.

Q14

Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is — start with any number; if the number is even, take half of it; if the number is odd, multiply it by $-3$ and add $1$ ; repeat. An example sequence is shown below.

Try this with different starting numbers: $(-21)$ , $(-6)$ , and so on. Describe the patterns you observe.

Q15

In a test, $(+4)$ marks are given for every correct answer and $(-2)$ marks are given for every incorrect answer.

(a) Anita answered all the questions in the test. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in the test?

(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?

Q16

Pick the pattern — find the operations done by the machine shown below.

Q17

Imagine you're in a place where the temperature drops by $5^\circ\text{C}$ each hour. If the temperature is currently at $8^\circ\text{C}$ , write an expression which denotes the temperature after 4 hours.

Q18

Find 3 consecutive numbers with a product of (a) $-6$ , (b) $120$ .

Q19

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?

Q20

Find the values of:

(a) $(32 \times (-18)) \div ((-36))$

(b) $(32) \div ((-36) \times (-18))$

(d) $(280 \times (-7)) \div ((-8) \times (-35))$

Q21

Arrange the expressions given below in increasing order:

(a) $(-348) + (-1064)$

(b) $(-348) - (-1064)$

(d) $(-348) \times (-1064)$

(e) $348 \times (-1064)$

(f) $348 \times 964$

Q22

Given that $(-548) \times 972 = -532656$ , write the values of:

(a) $(-547) \times 972$

(b) $(-548) \times 971$

Q23

Given that $207 \times (-33 + 7) = -5382$ , write the value of $-207 \times (33 - 7) = \text{___}?$

Q24

Use the numbers $3, -2, 5, -6$ exactly once and the operations '+', '-', and ' $\times$ ' 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

Q25

Fill in the blanks in at least 5 different ways with integers:

(a) $\square + \square \times \square = -36$

(b) $(\square - \square) \times \square = 12$