Aimathic – Integers and opposites

Integers and opposites

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Difficulty: 1 – 5

5225516

Write each banking amount as a signed number using $+$ or $-$. a) A credit of $\$75$ b) A withdrawal of $\$40$ c) An account balance of $\$12$ d) A debt of $\$100$

Hints

- Decide whether each amount adds to or takes away from what is available. - Does the situation represent a value greater than or less than zero? - Which sign represents money you have, and which sign represents money you owe?

Solution

1. A credit adds money, so it is represented by $+\$75$. 2. A withdrawal removes money, so it is represented by $-\$40$. 3. Money available in the account is represented by $+\$12$. 4. A debt is represented by $-\$100$.

Answer

a) $+\$75$ b) $-\$40$ c) $+\$12$ d) $-\$100$

5225596

Write each quantity as an integer with the appropriate sign. a) A desert basin is $430\,\text{m}$ below sea level. b) A mountain summit is $2962\,\text{m}$ above sea level. c) A research submersible is $250\,\text{m}$ below the ocean surface. d) A freezer is $18$ degrees Celsius below $0\,^\circ\text{C}$.

Hints

- Identify the zero reference point in each situation. - Words such as “below” indicate one direction from zero. - Words such as “above” indicate the opposite direction.

Solution

1. Quantities below a reference level are negative, and quantities above a reference level are positive. 2. The basin is represented by $-430$. 3. The mountain summit is represented by $+2962$. 4. The submersible is represented by $-250$. 5. The freezer temperature is represented by $-18$.

Answer

a) $-430$ b) $+2962$ c) $-250$ d) $-18$

5225816

An elevation of $-45\,\text{m}$ means a location is $45\,\text{m}$ below sea level. What does an elevation of $+2962\,\text{m}$ mean? Also explain what $0\,\text{m}$ represents in this context.

Hints

- What is the opposite of “below sea level”? - Picture a vertical number line through the water. - Where would the water surface be on that number line?

Solution

1. Negative elevations are below sea level, so positive elevations are above sea level. 2. An elevation of $+2962\,\text{m}$ means the location is $2962\,\text{m}$ above sea level. 3. An elevation of $0\,\text{m}$ is the reference point: sea level.

Answer

An elevation of $+2962\,\text{m}$ is $2962\,\text{m}$ above sea level. The value $0\,\text{m}$ represents sea level.

5103836

Decide whether each statement is true or false. Briefly justify your answer or give a counterexample. a) Every integer is a rational number. b) A fraction can never have a positive whole-number value. c) Every rational number corresponds to exactly one point on the number line.

Hints

- An integer can be written over a denominator of $1$. - Look for a fraction that simplifies to a positive whole number. - Think about how numbers are represented as points on a number line.

Solution

1. Statement a) is true. Any integer $z$ can be written as $\frac{z}{1}$, so it is rational. 2. Statement b) is false. For example, $\frac{8}{2} = 4$, which is a positive whole number. 3. Statement c) is true. Every rational number has one unique location on the number line.

Answer

a) True b) False; for example, $\frac{8}{2} = 4$ c) True

5103936

Imagine a Venn diagram with one circle for integers and another for rational numbers. 1. Which circle must lie completely inside the other? 2. Where should $\frac{12}{4}$ be placed: in the rational-only region or in the region for numbers that are both rational and integers? Explain.

Hints

- Every integer can be written as a fraction with denominator $1$. - Think of a rational number that is not an integer. - Simplify $\frac{12}{4}$ before placing it.

Solution

1. Every integer can be written as $\frac{n}{1}$, so every integer is rational. However, some rational numbers, such as $\frac{1}{2}$, are not integers. 2. Therefore, the integer circle lies completely inside the rational-number circle. 3. Evaluate $\frac{12}{4} = 3$. 4. Since $3$ is an integer, $\frac{12}{4}$ belongs in the integer region, which is also inside the rational-number region.

Answer

1. The integer circle lies completely inside the rational-number circle. 2. $\frac{12}{4}$ belongs in the region for both integers and rational numbers because $\frac{12}{4} = 3$.

5114466

Classify each number using the smallest category that contains it: positive whole number, integer, or rational number that is not an integer. $4.5$, $-12$, $\frac{20}{4}$, $0$, $-0.75$, $120\%$

Hints

- Simplify or convert each number before classifying it. - Positive whole numbers are $1, 2, 3, \ldots$. - Integers include zero and negative whole numbers. - Rational numbers can be written as a ratio of integers.

Solution

1. The value $4.5$ is rational but not an integer. 2. The value $-12$ is an integer but not a positive whole number. 3. Simplify $\frac{20}{4} = 5$, which is a positive whole number. 4. The value $0$ is an integer but not a positive whole number. 5. The value $-0.75$ is rational but not an integer. 6. Convert $120\% = 1.2 = \frac{6}{5}$, which is rational but not an integer.

Answer

$4.5$: rational noninteger $-12$: integer $\frac{20}{4}$: positive whole number $0$: integer $-0.75$: rational noninteger $120\%$: rational noninteger

5225676

Positive and negative numbers can represent opposite conditions or changes. Explain what each value means in its context. 1. A checking account balance is $-\$150$. 2. A store's net income is $-\$40$. 3. A diver's elevation is $-25\,\text{m}$ relative to sea level. 4. The temperature changes by $-8\,^\circ\text{C}$.

Hints

- Think about the opposite of a positive balance, a profit, an elevation above sea level, or an increase in temperature. - Identify what zero represents in each context. - Use the sign to determine the direction from zero.

Solution

1. A balance of $-\$150$ means the account is overdrawn by $\$150$. 2. Net income of $-\$40$ means the store had a loss of $\$40$. 3. An elevation of $-25\,\text{m}$ means the diver is $25\,\text{m}$ below sea level. 4. A change of $-8\,^\circ\text{C}$ means the temperature decreased by $8\,^\circ\text{C}$.

Answer

1. The account is overdrawn by $\$150$. 2. The store had a loss of $\$40$. 3. The diver is $25\,\text{m}$ below sea level. 4. The temperature decreased by $8\,^\circ\text{C}$.

5225776

The ideal temperature in a greenhouse is $20\,^\circ\text{C}$. A gardener records these temperatures: Monday: $23\,^\circ\text{C}$ Tuesday: $19\,^\circ\text{C}$ Wednesday: $20\,^\circ\text{C}$ Thursday: $17\,^\circ\text{C}$ For each day, write the deviation from the ideal temperature as a signed number. Use positive numbers for temperatures above the ideal and negative numbers for temperatures below the ideal.

Hints

- Compare each measured temperature with the target temperature. - Which sign represents a value below the target? - What does a deviation of zero mean?

Solution

1. Monday is $3$ degrees above the ideal, so the deviation is $+3\,^\circ\text{C}$. 2. Tuesday is $1$ degree below the ideal, so the deviation is $-1\,^\circ\text{C}$. 3. Wednesday equals the ideal, so the deviation is $0\,^\circ\text{C}$. 4. Thursday is $3$ degrees below the ideal, so the deviation is $-3\,^\circ\text{C}$.

Answer

Monday: $+3\,^\circ\text{C}$ Tuesday: $-1\,^\circ\text{C}$ Wednesday: $0\,^\circ\text{C}$ Thursday: $-3\,^\circ\text{C}$

5225826

In a quiz game, correct answers add points and incorrect answers subtract points. A score of $-10$ means a player is $10$ points below zero. a) What does a score of $+25$ mean? b) A player has a score of $-5$. She answers a question correctly and earns $5$ points. What is her new score, and what does it mean?

Hints

- If a negative score represents being below zero, what does a positive score represent? - What happens when a number and its opposite are combined? - Picture the scores on a number line.

Solution

1. A score of $+25$ means the player is $25$ points above zero. 2. Adding $5$ to $-5$ combines opposite numbers: $-5+5=0$. 3. A score of $0$ means the player is neither above nor below zero.

Answer

a) The player is $25$ points above zero. b) The new score is $0$, meaning the score is neither above nor below zero.

5103686

Answer each question about rational numbers and integers. a) Give a rational number that is not an integer and satisfies $-1 < x < -0.5$. b) Is this statement true or false? Explain: “Every positive whole number is an integer.” c) A student claims, “The sum of two integers is always a positive whole number.” Give a counterexample.

Hints

- A rational number can be written as a fraction of integers. - Think about how positive whole numbers fit inside the set of integers. - Use negative integers to produce a sum that is not positive.

Solution

1. For a), any noninteger rational number between $-1$ and $-0.5$ works. One example is $-0.75 = -\frac{3}{4}$. 2. For b), the statement is true. Every positive whole number is included among the integers. 3. For c), choose integers with a negative sum. For example, $-5 + 2 = -3$. Since $-3$ is not a positive whole number, the claim is false.

Answer

a) Answers will vary. One example is $-0.75$. b) True. Every positive whole number is an integer. c) Answers will vary. One counterexample is $-5 + 2 = -3$.

5103786

Let $x = -\frac{24}{6}$. a) Is $x$ an integer? Is it rational? b) Where is the opposite of $x$ located relative to zero on the number line? c) Give a rational number less than $x$ that is not an integer.

Hints

- Simplify the fraction first. - Every integer can be written as a fraction with denominator $1$. - Opposites are the same distance from zero on opposite sides. - Moving left on a number line gives smaller values.

Solution

1. Simplify: $x = -\frac{24}{6} = -4$. 2. The number $-4$ is an integer. Every integer is also rational because it can be written over $1$. 3. The opposite of $-4$ is $4$, which lies to the right of zero. 4. A rational noninteger less than $-4$ is $-4.5$. Many other answers are possible.

Answer

a) $x$ is an integer and a rational number. b) Its opposite is $4$, to the right of zero. c) Answers will vary. One example is $-4.5$.

5103846

Give one number that meets each condition. If the condition is impossible, explain why. a) The number is rational but not an integer. b) The number is an integer but not rational. c) The number is negative and rational, and its absolute value is a positive whole number.

Hints

- A rational number can be written as a ratio of integers. - Any integer can be written with denominator $1$. - Absolute value gives a number's nonnegative distance from zero.

Solution

1. For a), $\frac{1}{2}$ is rational but not an integer. 2. For b), no such number exists. Every integer $z$ is rational because $z = \frac{z}{1}$. 3. For c), $-5$ works. It is rational, and $\lvert-5\rvert = 5$, a positive whole number.

Answer

a) Answers will vary. One example is $\frac{1}{2}$. b) Impossible; every integer is rational. c) Answers will vary. One example is $-5$.

5105556

Let $S = \{-3, -0.5, 0, \frac{1}{4}, 2, 4.5\}$. Find all values $x$ in $S$ that meet each condition. a) $x$ is an integer. b) $x$ is rational but not an integer. c) $2x$ is a positive whole number.

Hints

- Check every value against each condition separately. - An integer has no fractional part. - For part c), calculate $2x$ for each value. - A positive whole number is $1, 2, 3, \ldots$.

Solution

1. For a), the integers in $S$ are $-3$, $0$, and $2$. 2. For b), the rational nonintegers are $-0.5$, $\frac{1}{4}$, and $4.5$. 3. For c), test each value: $2(-3) = -6$, $2(-0.5) = -1$, $2(0) = 0$, $2\left(\frac{1}{4}\right) = 0.5$, $2(2) = 4$, and $2(4.5) = 9$. 4. Only $x = 2$ and $x = 4.5$ give positive whole-number products.

Answer

a) $\{-3, 0, 2\}$ b) $\left\{-0.5, \frac{1}{4}, 4.5\right\}$ c) $\{2, 4.5\}$

5175546

Use the integers $-12,7,0,-5$. a) Write the opposite of each integer in the given order. b) Combine the original integers and their opposites, remove duplicates, and order the resulting numbers from least to greatest using $<$. c) List the pairs of different numbers that have the same absolute value.

Hints

- Taking an opposite changes the sign, except for zero. - Remove the repeated zero before ordering. - Different numbers with equal absolute value are opposites.

Solution

1. The opposites are $12,-7,0,5$. 2. The distinct original numbers and opposites are $-12,7,0,-5,12,-7,5$. In order, they are $-12<-7<-5<0<5<7<12$. 3. The pairs of different numbers with equal absolute value are $(-12, 12)$, $(-7, 7)$, and $(-5, 5)$. Zero has no different opposite.

Answer

a) $12,-7,0,5$ b) $-12<-7<-5<0<5<7<12$ c) $(-12, 12)$, $(-7, 7)$, and $(-5, 5)$

5217426

Use the numbers $12,-18,5,-3,0,-25$. a) Write the opposite of each number in the given order. b) Order the original numbers from least to greatest using $<$. c) Order the opposites from part a) from least to greatest.

Hints

- Opposite numbers are the same distance from zero on opposite sides. - On a number line, values increase from left to right. - Consider how taking opposites changes the order of a list.

Solution

1. Changing each number to its opposite gives $-12,18,-5,3,0,25$. 2. Ordering the original numbers gives $-25<-18<-3<0<5<12$. 3. Ordering the opposites gives $-12<-5<0<3<18<25$.

Answer

a) $-12,18,-5,3,0,25$ b) $-25<-18<-3<0<5<12$ c) $-12<-5<0<3<18<25$

5226696

Start with the positive whole numbers $1, 2, 3, \ldots$. a) Is the sum of any two positive whole numbers always a positive whole number? Is their product always a positive whole number? Give one example of each. b) Give an example showing that subtracting two positive whole numbers does not always produce a positive whole number. c) What numbers must be added to the positive whole numbers so that subtracting any two positive whole numbers always gives a number in the expanded set?

Hints

- Think about whether adding or multiplying counting numbers can take you outside the counting numbers. - Try subtracting a larger positive whole number from a smaller one. - Include the result of subtracting equal numbers and the numbers found to the left of $0$ on a number line.

Solution

1. Addition and multiplication are closed for positive whole numbers. For example, $4 + 7 = 11$ and $3 \times 5 = 15$, and both results are positive whole numbers. 2. Subtraction is not closed for positive whole numbers. For example, $3 - 8 = -5$, which is not positive. Also, subtracting a number from itself gives $0$, which is not positive. 3. Add $0$ and the negative integers $-1, -2, -3, \ldots$. Together with the positive whole numbers, these form the integers.

Answer

a) Yes. For example, $2 + 3 = 5$ and $2 \times 3 = 6$. b) Answers will vary. One example is $2 - 5 = -3$, which is not a positive whole number. c) Add $0$ and all negative integers.

5117276

Find one value for $\square$ that makes each condition true. a) $-\frac{1}{2} + \square$ is a positive whole number. b) $\square \times (-3)$ is a positive integer. c) $2.4 - \square$ is an integer that is not positive.

Hints

- For a), think of a value that moves $-0.5$ to a positive whole number. - For b), use the sign rules for multiplication. - For c), aim for zero or a negative integer.

Solution

1. For a), choose $\square = 1.5$. Then $-0.5 + 1.5 = 1$, a positive whole number. 2. For b), the missing value must be negative so the product is positive. One choice is $\square = -1$, because $(-1) \times (-3) = 3$. 3. For c), choose $\square = 2.4$. Then $2.4 - 2.4 = 0$, an integer that is not positive. 4. Other values may also satisfy each condition.

Answer

Answers will vary. One possible set is: a) $1.5$ b) $-1$ c) $2.4$

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Integers and opposites

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