Practice Questions

Describe the rule for multiplying two integers with different signs. Provide one example.

A student claims that for any two integers $a$ and $b$, the expression $a - b$ is always equal to $b - a$. Critique this statement and provide a counterexample to justify your position.

In a quiz, +5 marks are awarded for a correct answer and -3 marks for an incorrect answer. Rohan answers 12 questions correctly and 8 questions incorrectly. Calculate his final score.

Define the term 'additive inverse' of an integer and give an example.

Solve for the integer that replaces the blank: $(-84) \div$ ___ $= 7$.

What is the result when any integer 'a' is multiplied by $-1$?

Fill in the blank: The product of two negative integers is always a ________ integer.

Calculate the product: $(-6) \times (-5) \times (-2)$

Justify that subtraction is not associative for integers by creating a numerical example that demonstrates $(a - b) - c \neq a - (b - c)$.

Without performing any calculation, justify the sign (positive or negative) of the final product of the expression: $(-2) \times 5 \times (-11) \times 8 \times (-3)$.

Explain how to find the sum of two integers with opposite signs, for example, $(+8) + (-12)$, using the concept of a number line.

Formulate a general rule (or formula) to find two integers, $x$ and $y$, if their sum $S$ and their difference $D$ are known (assume $x > y$). Justify your rule algebraically. Then, use your rule to find the two integers whose sum is $-8$ and whose difference is $14$.

State the rule for the sign of the quotient when a positive integer is divided by a negative integer.

Recall and state the Commutative Property of multiplication for any two integers $a$ and $b$.

Summarize Brahmagupta's rules for the multiplication of 'fortunes' (positive numbers) and 'debts' (negative numbers).

Explain why subtracting a positive integer is equivalent to adding its negative counterpart. Use the example $(+7) - (+18)$.

Identify and explain the property of integer multiplication shown in the statement: $(-2) \times (5 + (-3)) = ((-2) \times 5) + ((-2) \times (-3))$

Describe the process of multiplying $(-2) \times 4$ using the token model. Explain the role of 'zero pairs' in this process.

List and explain the Associative and Distributive properties of multiplication for integers. For each property, provide a unique numerical example that includes at least one negative integer.

Simplify the following expression using the order of operations: (-125) ÷ 5 + [28 - (3 × 11)]

The temperature of a freezer is set to $-18^{\circ}$C. Due to a power failure, it starts to rise at a rate of $3^{\circ}$C per hour. Calculate the temperature inside the freezer after 5 hours.

A merchant gains ₹12 on selling one bag of premium rice and loses ₹7 on selling one bag of regular rice. (a) If he sells 250 bags of premium rice and 400 bags of regular rice in a month, calculate his overall profit or loss. (b) How many bags of premium rice must he sell to have neither profit nor loss, if he sells 540 bags of regular rice?

A submarine is positioned at 450 meters below sea level. It ascends 175 meters, then descends 220 meters, and finally ascends 300 meters. Calculate its final position relative to sea level.

Formulate an expression using three distinct integers from the set $\{-5, -2, 3, 4\}$ and at least two different operations ($+, -, \times, \div$) that evaluates to exactly $-14$.

Justify, using the concept of multiplication, why the division of any non-zero integer $a$ by $0$ is undefined.

Critique the following statement: "If the product of three integers is negative, then all three integers must be negative." Justify your critique with a counterexample.

Create a word problem about a deep-sea explorer's journey. The solution must involve the multiplication of a positive integer with a negative integer, followed by an addition to another negative integer. The final depth of the explorer must be $-350$ meters.

Two students, Aryan and Ben, evaluate the expression $(-8) \times [(-5) + 2]$. Aryan's answer is $56$, and Ben's answer is $24$. Evaluate both answers, justify who is correct, and formulate an explanation of the error made by the incorrect student, referencing the distributive property.

Observe the pattern: $(-3)^1 = -3$, $(-3)^2 = 9$, $(-3)^3 = -27$, $(-3)^4 = 81$. Formulate a general rule to determine the sign of the result of $(-3)^n$, where $n$ is any positive integer. Justify your rule.

Describe how to model the multiplication $3 \times (-2)$ using tokens, where red tokens represent negative integers and green tokens represent positive integers.

A diver is at a depth of 15 meters below sea level. If he descends another 20 meters, what is his new position? Represent the answer as an integer.

Apply the distributive property to calculate $(-15) \times 98$ in a simpler way.

An elevator descends into a mine shaft at a rate of 6 meters per minute. If it starts from 20 meters above ground level, calculate its position after 15 minutes.

Find a pair of integers whose product is $-48$ and whose difference is $16$.

Explain the pattern that emerges in the products when a positive integer, for example 4, is multiplied by a sequence of integers decreasing by 1, from 3 down to -3. Describe how the product changes at each step.

List the four possible sign combinations for the division of two non-zero integers and state the sign of the result for each case.

Design a game where a player starts at position 0 on a number line. In each of 3 turns, the player performs two actions: first, they roll a standard six-sided die (outcome $R$); second, they draw a card from a deck containing cards labeled 'add 2' and 'subtract 5'. The move for the turn is calculated as $R \times ( ext{card value})$. Formulate a strategy and propose one possible sequence of die rolls and card draws to land exactly on the number $-21$.

Create a $3 \times 3$ grid and fill it with nine distinct integers such that the product of the integers in each row and each column is equal to $-60$. Justify that your created grid meets all the conditions.

Analyze and determine the sign (positive or negative) of the final product if we multiply 10 negative integers and 5 positive integers.

For the integers $a = -4$, $b = 10$, and $c = -5$, demonstrate that multiplication is associative and distributive over addition. (a) Associative Property: $(a \times b) \times c = a \times (b \times c)$ (b) Distributive Property: $a \times (b + c) = (a \times b) + (a \times c)$

The temperature of a chemical is $20^\circ\text{C}$. A cooling process lowers its temperature by $4^\circ\text{C}$ per minute. A heating process raises its temperature by $5^\circ\text{C}$ per minute. You can only apply one process at a time for whole minutes. Design and justify a sequence of cooling and heating operations, lasting a total of exactly 8 minutes, to bring the final temperature to exactly $-7^\circ\text{C}$.

Using the integers $-4, -2, 3, 5$ each exactly once, and the operations $+, -, \times$ each exactly once, create expressions that result in: (a) The largest possible value. (b) The smallest possible value. Show the expressions and the calculations.

Design a scoring system for a quiz with 20 questions. Assign integer values for correct answers, incorrect answers, and unattempted questions. Propose a scheme where a student who answers 12 questions correctly, 5 incorrectly, and leaves 3 unattempted scores exactly $37$ marks.

Evaluate the statement: "The product of any two integers is always greater than their sum." Justify your conclusion by proposing examples where the statement is true and counterexamples where it is false.

Analyze the following pattern and determine the next two terms: $12, -36, 108, -324, ...$