Operations with Integers | FIO

Question 24

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

Solution

We need to arrange the numbers $3, -2, 5, -6$ and the operations '+', '-', 'x' to find the largest and smallest possible results. Each number and operation must be used exactly once.

Step 1 — Finding the maximum possible value

To get the largest possible positive number, we should multiply two numbers. These numbers should either both be large positive or both be large negative. We have the numbers $3, -2, 5, -6$ . Let us try to make a large negative number using three numbers. Then we will multiply it by the remaining negative number. Let us choose $-6$ as the multiplier. We need to make the expression using $3, -2, 5$ as negative as possible. We will use one '+' and one '-' operation. Consider the expression $[(-2) - (3 + 5)]$ . First, we add $3$ and $5$ .

$3 + 5 = \mathbf{8}$

Next, we subtract this sum from $-2$ .

$-2 - 8 = \mathbf{-10}$

Now, we multiply this result by the remaining number, which is $-6$ .

$(-10) \text{ x } (-6)$

$= \mathbf{60}$

This gives the maximum possible value.

$\boxed{\text{Maximum value} = 60}$

Diagram 1

Step 2 — Finding the minimum possible value

To get the smallest possible negative number, we should multiply a large positive number by a large negative number. We have the numbers $3, -2, 5, -6$ . Let us choose $-6$ as the negative multiplier. It has the largest negative value. We need to make the expression using $3, -2, 5$ as positive and as large as possible. We will use one '+' and one '-' operation. Consider the expression $[3 - (-2) + 5]$ . First, we subtract $-2$ from $3$ . Subtracting a negative number is the same as adding its positive.

$3 - (-2) = 3 + 2 = \mathbf{5}$

Next, we add $5$ to this result.

$5 + 5 = \mathbf{10}$

Now, we multiply this positive result by the remaining number, which is $-6$ .

$10 \text{ x } (-6)$

$= \mathbf{-60}$

This gives the minimum possible value.

$\boxed{\text{Minimum value} = -60}$

Diagram 2

Answer

(a) The maximum possible value is 60. (b) The minimum possible value is -60.

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) $(-7) \times 4 \times (-1)$ (c) $(-2) \times (-1) \times (-5) \times (-3)$

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°C each hour. If the temperature is currently at 8°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 $-207 \times (33 - 7) =\_\_\_?$

Q24

Use the numbers $3, -2, 5, -6$ exactly once and the operations '+', '-', and 'x' 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) $(\square - \square) \times \square = 12$