Equations, Inequalities & Formulae — Free IB MYP3 Mathematics Questions

1QuestionInterpreting Inequalities in Real-Life ScenariosConcept Practice

2 marks~3 minCriterion B

The table below shows four real-life scenarios and their corresponding inequality symbols and circle types on a number line.

Scenario: "At least 5 people"	symbol: $\geq$	circle: closed
Scenario: "Fewer than 3 hours"	symbol: $<$	circle: open
Scenario: "No more than 10 kg"	symbol: $\leq$	circle: closed
Scenario: "More than 2 km"	symbol: $>$	circle: open

a

Describe the pattern that links the inequality symbol to the type of circle used. [1]

b

Apply this pattern to the scenario "Less than 7 dollars" and state the circle type required. [1]

Solutions

2QuestionInterpreting Inequalities in Real-Life ScenariosConcept Practice

2 marks~3 minCriterion A

A rectangle has side lengths

(x + 3)

cm and

(x - 1)

cm. Its perimeter must be at least 20 cm.

a

Write an inequality to represent this condition and solve it for

x

. [1]

b

Hence, calculate the smallest possible integer length of the shorter side,

(x - 1)

cm. [1]

Solutions

3QuestionSolving Problems Involving Age and TimeConcept Practice

2 marks~3 minCriterion B

Tom is 10 years old, Anna is 6 years old, and Sam is 4 years old.

a

Complete the table of ages.

Today — Tom: 10, Anna: 6, Sam: 4

In 2 years — Tom: ___, Anna: ___, Sam: ___

In 5 years — Tom: ___, Anna: ___, Sam: ___ [1]

b

Using your completed table, explain whether the age difference between Tom and Anna changes over time, and state the general rule about age differences. [1]

Solutions

4QuestionChecking Solutions in the Context of the ProblemConcept Practice

2 marks~3 minCriterion A

A rectangle has side lengths of

2x + 3

cm and

x + 5

cm. Its perimeter is 46 cm.

a

Calculate the value of

x

. [1]

b

Justify whether your value of

x

gives valid side lengths for this rectangle. [1]

Solutions

5QuestionApplying the Speed Distance Time FormulaConcept Practice

2 marks~3 minCriterion A

A car travels at a constant speed of

75

km/h for

3

hours.

\text{Distance} = \text{Speed} \times \text{Time}

Calculate the total distance travelled by the car. [2]

Solutions

6QuestionSolving Equations with Decimals and FractionsConcept Practice

2 marks~3 minCriterion B

Solve each equation for

x

.

a

Calculate the value of

x

in each equation. [1]

x + 0.3 = 0.7 \qquad x + 0.25 = 0.75 \qquad x + 0.125 = 0.875 \qquad x + 0.1 = 0.9

b

Describe the pattern connecting the decimal added to

x

and the solution for

x

in each equation. [1]

Solutions

7QuestionSolving One-Step Addition and Subtraction EquationsConcept Practice

3 marks~5 minCriterion C

The bar model below represents an equation.

a

Describe what the bar model shows about the relationship between the whole and its parts. [1]

b

Explain how you would use an inverse operation to isolate

x

. [1]

c

A student claims that

x = 21

. Calculate the correct value of

x

and state whether the student is right or wrong. [1]

Solutions

8QuestionSolving Equations with Decimals and FractionsConcept Practice

2 marks~3 minCriterion A

A rectangle has side lengths of

2.5x

cm and

4.2

cm. Its perimeter is

20.6

cm.

The perimeter of a rectangle is given by

P = 2(\text{length} + \text{width})

.

Calculate the value of

x

. [2]

Solutions

9QuestionExpanding Brackets Before SolvingConcept Practice

2 marks~3 minCriterion B

Each expression below has been expanded using the distributive property.

2(x + 1) = 2x + 2

3(x + 2) = 3x + 6

4(x + 3) = 4x + 12

5(x + 4) = 5x + 20

a

Describe the pattern you see in how each expression is expanded. [1]

b

Apply this pattern to expand

6(x + 5)

. Show your working. [1]

Solutions

10QuestionSolving Equations Involving FractionsConcept Practice

2 marks~3 minCriterion A

A triangle has angles measuring

x°

,

\dfrac{x}{2}°

, and

30°

.

(a) Write an equation using the angle sum property of a triangle. Calculate the value of

x

. [2]

Solutions

11QuestionSolving Equations with Variables on Both SidesConcept Practice

3 marks~5 minCriterion C

The graph shows the lines

y = 2x + 1

and

y = x + 5

, which intersect at the point

(4, 9)

.

a

Calculate the value of each side of the equation

2x + 1 = x + 5

when

x = 4

. [1]

b

Explain what the equation

2x + 1 = x + 5

is asking you to find. [1]

c

Interpret what the

x

-coordinate of the intersection point represents in relation to solving the equation

2x + 1 = x + 5

. [1]

Solutions

12QuestionSolving Two-Step InequalitiesAssessment Practice

3 marks~5 minCriterion C

The graph shows the line

y = 3x - 7

and the horizontal line

y = 8

. The two lines intersect at the point

(5, 8)

.

a

State what the intersection point

(5, 8)

tells you about the equation

3x - 7 = 8

. [1]

b

Explain how the graph shows the solution to the inequality

3x - 7 \leq 8

. [1]

c

Describe the complete solution set of

3x - 7 \leq 8

and justify why

x = 6

is not part of it. [1]

Solutions

13QuestionGraphing Inequalities on a Number LineAssessment Practice

5 marks~8 minCriterion D

A student models a car's speed using the inequality

v \leq 60

km/h, where

v

is the speed in km/h. The number line below shows this model.

During the same time interval, the car's actual speed ranges from 55 km/h to 65 km/h.

a

Interpret what the inequality

v \leq 60

km/h means in this context. [1]

b

Explain whether the inequality model accurately reflects the car's actual speed. [2]

c

Explain one limitation of using a single inequality to represent the car's speed. [2]

Solutions

14QuestionChecking Solutions in the Context of the ProblemAssessment Practice

4 marks~6 minCriterion D

A school bake sale has a fixed booth fee of 30 dollars and a cost of 2 dollars per cupcake. The total cost

C

(in dollars) for making

x

cupcakes is

C = 2x + 30

. Each cupcake sells for 5 dollars.

a

Write an equation for the value of

x

that gives a profit of exactly 100 dollars. [1]

b

Calculate the value of

x

. [1]

c

Analyse whether your answer to part (b) is reasonable for this situation. Support your reasoning with calculations. [2]

Solutions

15QuestionTranslating Word Problems into EquationsAssessment Practice

6 marks~9 minCriterion C

A basketball court charges 5 dollars for the first hour and 8 dollars for each additional hour. A student models the total cost for

x

hours with the equation

5x + 20 = 110

, which incorrectly includes a flat booking fee of 20 dollars.

a

Calculate the number of hours given by the student's model when the total cost is 110 dollars. [1]

b

Write the correct piecewise function

C(x)

for the actual total cost of

x

hours, where

x \geq 1

. [2]

c

Explain why the student's single linear equation does not accurately represent the actual pricing structure, using values from both models to support your answer. [3]

Solutions

16QuestionUsing Volume and Surface Area FormulasAssessment Practice

3 marks~5 minCriterion B

The table below shows the volume

V

(in cm³) of a cube with side length

s

(in cm).

$s$ (cm)	1	2	3	4
$V$ (cm³)	1	8	27	64

These points are plotted on a graph with a smooth curve connecting them.

a

Apply the formula

V = s^3

to calculate the volume of a cube with side length

s = 5

cm. [1]

b

Explain why the graph of

V = s^3

is a curve and not a straight line. [1]

c

The increases in volume between consecutive values of

s

in the table are 7, 19, and 37. Analyse what these increasing differences tell you about the rate at which volume grows as side length increases. [1]

Solutions

17QuestionUsing the Formula for Area of ShapesAssessment Practice

5 marks~8 minCriterion C

A gardener designs an L-shaped flower bed made from two rectangles. The left rectangle measures

3 \text{ m} \times 5 \text{ m}

and the right rectangle measures

4 \text{ m} \times 2 \text{ m}

. Their top edges are aligned.

a

Describe how dividing the L-shape into two rectangles allows you to find the total area. [1]

b

Calculate the area of each rectangle and the total area of the flower bed. [2]

c

A second flower bed is a single rectangle measuring

7 \text{ m} \times 5 \text{ m}

. The gardener says the L-shaped bed uses less than half the area of the rectangular bed. Justify whether the gardener is correct. [2]

Solutions

18QuestionApplying the Speed Distance Time FormulaAssessment Practice

6 marks~9 minCriterion D

A delivery company uses the formula

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

to schedule drivers in a city. Driver A travels 120 km, Driver B travels 90 km, and Driver C travels 60 km. All three drive at an average speed of 60 km/h through city routes with traffic lights, speed bumps, and variable traffic.

a

Calculate the theoretical time, in hours, for each driver. [2]

b

Explain why the constant-speed assumption is unrealistic for city driving. Give two specific examples from the scenario. [2]

c

A manager notices that Driver A consistently arrives 30 minutes later than the formula predicts. Calculate Driver A's actual average speed, then analyse what this tells you about the reliability of the formula for scheduling. [2]

Solutions

19QuestionSolving Equations with Decimals and FractionsAssessment Practice

5 marks~8 minCriterion D

A student has a total budget of 45.00 dollars to spend at a fair. The entry fee is 12.50 dollars and each ride costs 0.85 dollars. The student uses the equation

0.85x + 12.50 = 45.00

, where

x

is the number of rides.

a

Calculate the number of rides the student can afford using this equation. Show all working. [2]

b

Explain why the model may overestimate the number of affordable rides if a 10% sales tax is applied to each ride token. [1]

c

Calculate the number of rides the student can afford when the 10% sales tax is included. Show all working and compare this with your answer to part (a). [2]

Solutions

20QuestionClearing Fractions in EquationsAssessment Practice

6 marks~9 minCriterion D

A student receives a monthly allowance of 500 dollars. They plan to save

x

dollars and spend the rest. The situation is modelled by the equation

\frac{x}{4} + \frac{500 - x}{3} = 150

a

Calculate the value of

x

by first clearing the fractions. [2]

b

Explain one assumption the model makes about the student's monthly spending, and describe how this assumption could limit the model's accuracy. [2]

c

The student uses this model over 12 months. Analyse how a rounding error of 0.50 dollars per month in the savings amount could affect the student's total annual budget prediction. [2]

Solutions

Equations, Inequalities & Formulae — Free MYP3 Mathematics Practice Questions