MATHEMATICS - EQUATIONS WITH VARIABLES ON BOTH SIDES

Equations With Variables On Both Sides Worksheet

  • 10 questions
  • Understand that a situation can be represented either in words or as an equation. Move between the two representations and solve the equation (integer or fractional coefficients, unknown on either or both sides).
  • Difficulty: Medium
  • Board: CAIE
  • Attempt online on the examination portal

Here are the first 3 questions. Click “View all questions” to see the complete assessment.

Q1.

Plan A charges 12 dollars plus 4 dollars per visit. Plan B charges 30 dollars plus 2 dollars per visit. If v is the number of visits, which equation represents equal total costs, and how many visits make the costs equal?

  1. A) 4v − 12 = 30 − 2v, so v = 7
  2. B) 12 + 4v = 30 + 2v, so v = 21
  3. C) 12 + 4v = 30 + 2v, so v = 9
  4. D) 12 + 2v = 30 + 4v, so v = 9

Q2.

Maya uses 1/2x + 7 metres of ribbon. Liam uses 3/4x − 2 metres. They use equal lengths. Which equation represents the situation, and what is x?

  1. A) 1/2x − 7 = 3/4x + 2, so x = − 36
  2. B) 3/4x + 7 = 1/2x − 2, so x = − 36
  3. C) 1/2x + 7 = 3/4x − 2, so x = 36
  4. D) 1/2x + 7 = 3/4x + 2, so x = 20

Q3.

Five less than three times a number is equal to one more than twice the number. If the number is n, which equation and solution represent this statement?

  1. A) 3n − 5 = 2n + 1, so n = 6
  2. B) 5 − 3n = 2n + 1, so n = 4/5
  3. C) 3n + 5 = 2n − 1, so n = − 6
  4. D) 3(n − 5) = 2(n + 1), so n = 17

Q4.

Two rectangles have equal perimeters. Rectangle A has length (x + 2) cm and width 4 cm. Rectangle B has length (2x − 3) cm and width 2 cm. Which equation represents the equal perimeters, and what is x?

  1. A) 2(x + 2) + 2(4) = 2(2x − 3) + 2(2), so x = 7
  2. B) 2(x + 2) + 4 = 2(2x − 3) + 2, so x = 5
  3. C) 2(x + 2) + 2(4) = 2(2x − 3) + 2(2), so x = 14
  4. D) (x + 2) + 2(4) = (2x − 3) + 2(2), so x = 9

Q5.

Tank A contains (3/5x + 8 ) litres of water. Tank B contains (1/4x + 22 ) litres. The tanks contain equal amounts. Which equation represents this situation, and what is x?

  1. A) 3/5x + 8 = 1/4x + 22, so x = 14
  2. B) 3/5x + 8 = 1/4x + 22, so x = 40
  3. C) 3/5x + 22 = 1/4x + 8, so x = − 40
  4. D) 3/5x − 8 = 1/4x + 22, so x = 600/7

Q6.

Taxi P charges a starting fee of 6 dollars and 5/2 dollars per kilometre. Taxi Q charges a starting fee of 18 dollars and 3/2 dollars per kilometre. If k is the distance travelled, which equation represents equal fares, and at what distance are they equal?

  1. A) 6 + 3/2k = 18 + 5/2k, so k = 12 km
  2. B) 6 + 5/2k = 18 + 3/2k, so k = 12 km
  3. C) 6 + 5/2k = 18 + 3/2k, so k = 6 km
  4. D) 5/2(k + 6) = 3/2(k + 18), so k = 12 km

Q7.

Subtracting 4 from half a number gives the same result as adding 5 to one third of the number. If the number is y, which equation and solution represent the statement?

  1. A) 2y − 4 = 3y + 5, so y = − 9
  2. B) y/2 + 4 = y/3 − 5, so y = − 54
  3. C) y/2 − 4 = y/3 + 5, so y = 54
  4. D) (y − 4)/(2) = (y + 5)/(3), so y = 22

Q8.

Zara starts with 4 bags containing c counters each and then gives away 7 counters. Imran starts with 2 identical bags and then receives 13 counters. They finish with the same number of counters. Which equation represents the situation, and how many counters are in each bag?

  1. A) 4(c − 7) = 2(c + 13), so c = 27
  2. B) 4c − 7 = 2c + 13, so c = 10
  3. C) 4c − 13 = 2c + 7, so c = 10
  4. D) 4c + 7 = 2c − 13, so c = − 10

Q9.

The balance scale in the figure is level. Each bag has the same mass, represented by m. Which equation represents the balance, and what is the mass of one bag?

Diagram for question 9: The balance scale in the figure is level. Each bag has the same mass, represented by m. Which equation…
  1. A) 3m + 2 = 10m, so m = 2/7 kg
  2. B) 3m + 2 = m + 10, so m = 4 kg
  3. C) 3m + 10 = m + 2, so m = − 4 kg
  4. D) 3(m + 2) = m + 10, so m = 2 kg

Q10.

The figure shows two bars with the same total length. Which equation represents their equal lengths, and what is x?

Diagram for question 10: The figure shows two bars with the same total length. Which equation represents their equal lengths, and…
  1. A) x/3 + 2 = x/2 + 8, so x = − 36
  2. B) x/3 + 8 = x/2 + 2, so x = 6
  3. C) (x + 8)/(3) = (x + 2)/(2), so x = 10
  4. D) x/3 + 8 = x/2 + 2, so x = 36

What this assessment covers

Concepts Tested

  • equations with variables on both sides
  • Understand that a situation can be represented either in words or as an equation. Move between the two representations and solve the equation (integer or fractional coefficients, unknown on either or both sides).

Question Mix

  • Medium questions included

Skills you build

  • Understand

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