Grade 10 Maths · Algebra

Calculate with integers

Add, subtract, multiply and divide positive and negative whole numbers in the right order (for example −3 × −4 = 12, and (7 − 4)³ = 27).

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Question 1.1 · March 2022 · Provincial · 2 marks

Four options are provided as possible answers to the following questions. Choose the correct option.

1.1 If

k

is a negative single digit integer, which of the following is NOT a possible value of

2 - k

?(2)

A 3
B 4
C 10
D 12

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: calculate with integers

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3. Question 1.2.2 · March 2024 · Paper 1 · Provincial · 2 marks
1.2 Given: $x^{2} - x - 20 < 0$
1.2.2 Hence, or otherwise, determine the sum of all the integers satisfying the inequality $x^{2} - x - 20 < 0$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify integers in the intervalIntegers: $- 3; - 2; - 1; 0; 1; 2; 3; 4$
  1 mark · Write or draw a solution set
2. ✓
  Calculate sumSum $= - 3 - 2 - 1 + 0 + 1 + 2 + 3 + 4 = 4$
  1 mark · Calculate with integers
4. Question 4.3 · March 2023 · Provincial · 4 marks
$(1 - x)$ ; $(x + 2)$ and $(2 x - 5)$ are the first three (3) terms of an arithmetic sequence.
4.3 Hence, or otherwise, calculate $m$ if: $\sum_{n = 0}^{m - 1} (24 - 15 n) = S_{100} + 73320$ (4)
Show the memo (5 ticks)Memo
1. ✓
  Calculate RHS value $R H S = - 73350 + 73320 = - 30$
  1 mark · Calculate with integers
2. ✓
  Analyze Sigma seriesSeries terms: $n = 0 \to 24$ , $n = 1 \to 9$ , $n = 2 \to - 6$
  $a = 24, d = - 15$ . Number of terms is $m$ .
  0 marks · Read and write sigma notation
3. ✓
  Set up sum equation for m terms $\frac{m}{2} [2 (24) + (m - 1) (- 15)] = - 30$
  1 mark · Calculate an arithmetic series sum
4. ✓
  Form quadratic equation $m (48 - 15 m + 15) = - 60$
  $63 m - 15 m^{2} = - 60$
  $15 m^{2} - 63 m - 60 = 0$
  1 mark · Write a quadratic in standard form
5. ✓
  Solve for m $5 m^{2} - 21 m - 20 = 0$
  $(5 m + 4) (m - 5) = 0$
  $m = 5$ (since $m$ must be positive integer)
  1 mark · Solve a quadratic by factorising
5. Question 9.2 · March 2021 · Provincial · 6 marks
The diagram is the top view design of a new railway system. AF || BE and AC || GD. $\frac{A B}{BC} = \frac{4}{7}$ and $\frac{A G}{A F} = \frac{9}{17}$ .
9.2 If the straight line distance of the track from F to C is 374 km and it takes 50 hours to build one km of the track, determine the number of hours it will take to build the section from E to D.(6)
Show the memo (6 ticks)Memo
1. ✓
  $FE = 4 p$ and $EC = 7 p$ Let $FE = 4 p$ and $EC = 7 p$ (since $\frac{FE}{FC} = \frac{4}{11}$ )
  1 mark · Use a ratio, rate or proportion
2. ✓
  $p = 34$ $11 p = 374 ⟹ p = 34$
  $FE = 4 (34) = 136$ km
  1 mark · Solve a linear equation
3. ✓
  $F D = 8 m$ and $D C = 9 m$ Let $F D = 8 m$ and $D C = 9 m$ (since $\frac{C D}{D F} = \frac{9}{8}$ )
  1 mark · Use a ratio, rate or proportion
4. ✓
  $m = 22$ $17 m = 374 ⟹ m = 22$
  $C D = 9 (22) = 198$ km
  1 mark · Solve a linear equation
5. ✓
  $E D = 40$ km $E D = 374 - 4 p - 9 m = 374 - 136 - 198 = 40$ km
  1 mark · Calculate with integers
6. ✓
  Answer: 2 000 hoursTime $= 40 \times 50 = 2000$ hours
  1 mark · Use a ratio, rate or proportion
6. Question 1.1.2 · March 2018 · Provincial · 4 marks
1.1 Solve for x in each of the following:
1.1.2 $3^{x + 1} - 3^{x + 3} = - \frac{8}{27}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Factorise LHS $3^{x} (3^{1} - 3^{3}) = - \frac{8}{27}$
  1 mark · Take out a common power
2. ✓
  Simplify bracket $3^{x} (3 - 27) = - \frac{8}{27}$
  $3^{x} (- 24) = - \frac{8}{27}$
  1 mark · Calculate with integers
3. ✓
  Isolate base $3^{x} = \frac{- 8}{27} \div - 24$
  $3^{x} = \frac{1}{81}$
  1 mark · Solve a linear equation
4. ✓
  Solve for x $3^{x} = 3^{- 4}$
  $x = - 4$
  1 mark · Solve an exponential equation

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