Adding and Subtracting Integers: practice solutions, Grade 7

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Practice solutions Grade 7 : Adding and Subtracting Integers — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Writing opposites ★★★

The opposite flips the sign: \( -9 \), \( 14 \), \( 0 \) (zero is its own opposite), \( -2.5 \).

2 Absolute values ★★★

The absolute value is the distance from 0: \( |-8| = 8 \), \( |5| = 5 \), \( |0| = 0 \), \( |-12| = 12 \). Since \( |-7| = 7 \) and \( |4| = 4 \), \( |-7| \) is larger.

3 Quick sums ★★★

(a) Same signs: \( 3 + 9 = 12 \), so \( -12 \). (b) \( 8 - 5 = 3 \). (c) \( 10 - 6 = 4 \), positive. (d) Opposites: \( 0 \). (e) \( 15 - 4 = 11 \) with the sign of 15: \( -11 \).

4 Quick differences ★★★

(a) \( 6 + (-10) = -4 \). (b) \( 3 + 5 = 8 \). (c) \( -4 + (-7) = -11 \). (d) \( -2 + 9 = 7 \). (e) \( 0 + 6 = 6 \).

5 Morning warm-up ★★★

\( -8 + 15 = 7 \). Moving 15 right from \( -8 \) passes 0 after 8 steps and continues 7 more. The temperature is \( 7^\circ\text{F} \).

6 True or false? ★★★

(a) True: \( -3 \) is 3 units from 0. (b) True: \( -(-6) = 6 \). (c) False: signs differ, \( 5 - 2 = 3 \), so the sum is \( -3 \). (d) False: \( 4 - (-4) = 4 + 4 = 8 \).

7 Reading the number line ★★★

A is at \( -6 \), opposite 6, distance 6. B is at \( -2 \), opposite 2, distance 2. C is at 3, opposite \( -3 \), distance 3. D is at 5, opposite \( -5 \), distance 5.

8 Submarine dive ★★★

After rising: \( -45 + 18 = -27 \). After diving: \( -27 - 30 = -57 \). The submarine is at \( -57 \) ft. In meters: \( 57 \times 0.3048 = 17.37\ldots \approx 17.4 \) m, so about \( -17.4 \) m.

9 Distances ★★★

(a) \( |5 - (-7)| = 12 \). (b) \( |-2 - (-9)| = |7| = 7 \). (c) \( |11 - 3| = 8 \). (d) \( |0 - (-4)| = 4 \).

10 Bank account ★★★

\( 42 - 65 = -23 \), then \( -23 + 30 = 7 \), then \( 7 - 12 = -5 \). The balance is \( -5 \) dollars: the account is overdrawn by 5 dollars.

11 Missing numbers ★★★

(a) \( -2 - (-6) = 4 \). (b) \( -10 + 7 = -3 \). (c) \( 5 - 12 = -7 \), check \( 5 - (-7) = 12 \). (d) \( 1 + (-3) = -2 \), check \( -2 + 3 = 1 \).

12 Find the error ★★★

She treated the problem as \( -8 + 5 \). But \( -8 - 5 = -8 + (-5) \), and both numbers are negative, so we add the absolute values and keep the minus sign: \( -13 \).

13 Trivia game ★★★

Running totals: \( 15 \), \( 15 - 10 = 5 \), \( 5 + 5 = 10 \), \( 10 - 20 = -10 \), \( -10 + 10 = 0 \). The final score is 0. The lowest total, \( -10 \), came after round 4.

14 The hiker ★★★

Distance: \( 340 - (-120) = 340 + 120 = 460 \). She climbs 460 feet. Check: 120 ft up to sea level, then 340 ft more.

15 Rational numbers ★★★

(a) \( 4.75 - 2.5 = 2.25 \), positive. (b) \( -\dfrac{3}{4} + \dfrac{2}{4} = -\dfrac{1}{4} \). (c) \( 1.2 + 3.8 = 5 \). (d) \( -\dfrac{5}{6} + \dfrac{2}{6} = -\dfrac{3}{6} = -\dfrac{1}{2} \).

16 A long expression ★★★

Left to right: \( -12 + 9 = -3 \); \( -3 - (-4) = 1 \); \( 1 - 7 = -6 \). Check: rewrite as \( -12 + 9 + 4 + (-7) \). Positives: \( 9 + 4 = 13 \). Negatives: \( -12 - 7 = -19 \). Then \( 13 + (-19) = -6 \).

17 Two mystery integers ★★★

Let the integers be \( x \) and \( y \): \( x + y = -3 \) and \( x - y = 11 \). Adding both equations, \( 2x = 8 \), so \( x = 4 \). Then \( y = -3 - 4 = -7 \). Check: \( 4 + (-7) = -3 \) and \( 4 - (-7) = 11 \).

18 Points at a given distance ★★★

Move 9 right: \( -4 + 9 = 5 \). Move 9 left: \( -4 - 9 = -13 \). So Q is at 5 or at \( -13 \). Check: \( |5 - (-4)| = 9 \) and \( |-13 - (-4)| = |-9| = 9 \).

19 Weather extremes ★★★

(a) \( 5 - (-11) = 16 \), so it fell 16 degrees. (b) Warmest 14, coldest \( -11 \): \( 14 - (-11) = 25 \). The gap is 25 degrees Fahrenheit.

20 Always, sometimes, never ★★★

(a) Always: you move left twice, so you stay below 0. (b) Sometimes: \( 8 - 3 = 5 \) is positive but \( 3 - 8 = -5 \) is not. (c) Always: this is the additive inverse. (d) Sometimes: true for \( a = 6 \), false for \( a = -6 \) since \( |-6| = 6 \neq -6 \).

21 Elevator trip ★★★

(a) \( 2 \to 2 - 5 = -3 \to -3 + 9 = 6 \to 6 - 7 = -1 \to 8 \). (b) The last move is \( 8 - (-1) = 9 \) floors up. Total travel: \( 5 + 9 + 7 + 9 = 30 \) floors.

22 Swapping the order ★★★

\( a - b = -5 - 3 = -8 \) and \( b - a = 3 - (-5) = 8 \). They are opposites. Both have absolute value equal to the distance between \( a \) and \( b \) (8 units), but the walk goes in opposite directions, so the signs differ: \( b - a = -(a - b) \).