Algebra & Formulas | Quantitative Reasoning for UCAT PDF Download

Concept

Algebraic expressions are mathematical phrases combining variables (e.g., x, y), constants (e.g., 5, 10), and operations (e.g., +, ×). In UCAT, you’ll encounter expressions representing costs, quantities, or measurements. Key skills include:

• Simplifying: Combining like terms (e.g., 2x + 3x = 5x) to make expressions easier to handle.
• Expanding: Distributing terms across brackets (e.g., 3(x + 2) = 3x + 6) to prepare for solving or substitution.
• Factorizing: Extracting common factors (e.g., 4x + 6 = 2(2x + 3)) to simplify or interpret expressions.

These skills help you interpret word problems, such as total costs or quantities, and set up equations for further calculations.

Example 1: Simplifying Expressions

A clinic’s cost for x syringes is 4x + 7x + 15 - 2x. Simplify the expression.

Solution:

Combine like terms:

4x + 7x - 2x + 15 = (4x + 7x - 2x) + 15 = 9x + 15

Answer: 9x + 15

UCAT Tip: Group terms mentally: 4x + 7x = 11x, then 11x - 2x = 9x. If x = 2, check: 9 × 2 + 15 = 33, vs. original: (4 × 2) + (7 × 2) + 15 - (2 × 2) = 8 + 14 + 15 - 4 = 33.

Example 2: Expanding Brackets

Expand: 5(2x + 3) for a clinic’s cost of x kits plus a fixed fee.

Solution:

Distribute: 5(2x + 3) = 5 × 2x + 5 × 3 = 10x + 15

Answer: 10x + 15

UCAT Tip: Double-check distribution. Verify: if x = 1, 5(2 × 1 + 3) = 5 × 5 = 25, and 10 × 1 + 15 = 25.

Example 3: Factorizing

Factorize: 8x + 12 for a cost expression.

Solution:

Find the greatest common factor (GCF) of 8 and 12: 4

8x + 12 = 4(2x + 3)

Answer: 4(2x + 3)

UCAT Tip: Factorizing is rare in UCAT but useful for simplifying. Check: 4(2 × 2 + 3) = 4 × 7 = 28, vs. 8 × 2 + 12 = 28.

Concept

Linear equations are equations of the form ax + b = c, where x is the unknown. Solving involves isolating x using inverse operations (e.g., subtract to undo addition, divide to undo multiplication). UCAT uses linear equations to find quantities like the number of items or time taken. Steps:

1. Simplify both sides if needed (e.g., combine terms).
2. Isolate the variable term (e.g., subtract constants).
3. Solve for the variable (e.g., divide by the coefficient).
4. Verify by substituting back.

Equations may involve fractions, decimals, or multiple steps, but UCAT keeps them straightforward.

Example 1: Simple Linear Equation

A clinic spends 6x + 8 = 44 on x syringes. How many syringes were bought?

Solution:

6x + 8 = 44

Subtract 8: 6x = 44 - 8 = 36

Divide by 6: x = 36 ÷ 6 = 6

Answer: 6 syringes

UCAT Tip: Solve mentally for small numbers: 44 - 8 = 36, 36 ÷ 6 = 6. Check: 6 × 6 + 8 = 36 + 8 = 44.

Example 2: Equation with Fractions

Solve: x/5 + 4 = 9 for the number of kits x.

Solution:

x/5 + 4 = 9

Subtract 4: x/5 = 9 - 4 = 5

Multiply by 5: x = 5 × 5 = 25

Answer: 25 kits

UCAT Tip: Clear fractions by multiplying: multiply both sides by 5 to get x + 20 = 45, x = 25. Use the calculator for decimals if needed.

Example 3: Multi-Step Equation

Solve: 3(2x - 1) = 15 for x.

Solution:

Expand: 6x - 3 = 15

Add 3: 6x = 15 + 3 = 18

Divide by 6: x = 18 ÷ 6 = 3

Answer: x = 3

UCAT Tip: Write steps on the notebook to avoid errors. Check: 3(2 × 3 - 1) = 3 × 5 = 15.

Concept

Formulas are equations that define relationships between variables (e.g., speed = distance/time). UCAT tests your ability to:

• Substitute: Plug in known values to find unknowns.
• Rearrange: Manipulate formulas to solve for a specific variable.
• Apply: Use formulas in contexts like geometry, physics, or finance.

Common UCAT formulas include:

• Area of a rectangle: A = l × w
• Speed: s = d/t
• Volume of a cuboid: V = l × w × h
• Cost: Total = unit cost × quantity

Understanding units (e.g., km/h for speed) is crucial to avoid errors.

Example 1: Substitution

Find the area of a ward: A = l × w, where l = 10, w = 6.

Solution:

A = 10 × 6 = 60

Answer: 60 square units

UCAT Tip: Multiply mentally for small numbers. Check units: length × width = area.

Example 2: Rearranging

A car travels 180 km at 60 km/h. Find time: t = d/s.

Solution:

t = 180 ÷ 60 = 3

Answer: 3 hours

UCAT Tip: Use the calculator for division. Verify: 60 × 3 = 180 km.

Example 3: Complex Substitution

Find the cost: C = 2.5q + 10, where q = 8 (quantity of items).

Solution:

C = 2.5 × 8 + 10 = 20 + 10 = 30

Answer: 30

UCAT Tip: Break down: 2.5 × 8 = 20 (use calculator if needed). Check: 2.5 × 8 + 10 = 30.

Concept

Proportions compare ratios (e.g., 2/3 = 4/6). Variation describes relationships:

• Direct Variation: y = kx (e.g., cost increases with quantity).
• Inverse Variation: y = k/x (e.g., time decreases as workers increase).

UCAT uses proportions for scaling (e.g., costs, distances) and variation for rates or efficiencies. Steps:

1. Set up the proportion or variation equation.
2. Find the constant k if needed.
3. Solve for the unknown.

Example 1: Direct Proportion

If 4 masks cost 20, how much do 10 masks cost?

Solution:

Cost per mask: 20 ÷ 4 = 5

10 masks: 10 × 5 = 50

Answer: 50

UCAT Tip: Set up proportion: 4/20 = 10/x → x = (10 × 20) ÷ 4 = 50. Estimate: double 4 to 8 costs 40, add half for 2 more.

Example 2: Inverse Variation

If 5 workers complete a task in 8 hours, how long do 10 workers take?

Solution:

Total work: 5 × 8 = 40 worker-hours

Time for 10 workers: 40 ÷ 10 = 4 hours

Answer: 4 hours

UCAT Tip: Use inverse proportion: (5/10) × 8 = 0.5 × 8 = 4. Check: more workers, less time.

Example 3: Complex Proportion

If 3 liters of solution cost 45, how much do 7 liters cost?

Solution:

Cost per liter: 45 ÷ 3 = 15

7 liters: 7 × 15 = 105

Answer: 105

UCAT Tip: Proportion: 3/45 = 7/x → x = (7 × 45) ÷ 3 = 105. Use calculator for speed.

Concept

Inequalities (e.g., x < 5, x ≥ 2) describe ranges of values, often for constraints like budgets or quantities. Solve like equations, but:

• Flip the inequality sign when multiplying/dividing by a negative (e.g., -2x < 4 → x > -2).
• Interpret solutions in context (e.g., whole numbers for items).

UCAT uses inequalities for maximum/minimum values or feasibility checks.

Example 1: Simple Inequality

A budget allows 3x + 5 ≤ 35 for x kits. Find the maximum number of kits.

Solution:

3x + 5 ≤ 35

3x ≤ 35 - 5 = 30

x ≤ 30 ÷ 3 = 10

Since kits are whole numbers, x ≤ 10.

Answer: Maximum 10 kits

UCAT Tip: Check: 3 × 10 + 5 = 35 (within budget). Test 11: 3 × 11 + 5 = 38 (over budget).

Example 2: Negative Coefficient

Solve: -2x + 4 ≥ 10 for x.

Solution:

-2x + 4 ≥ 10

-2x ≥ 10 - 4 = 6

Divide by -2 (flip sign): x ≤ 6 ÷ -2 = -3

Answer: x ≤ -3

UCAT Tip: Rare in UCAT, but watch sign flips. Check: -2 × -3 + 4 = 6 + 4 = 10 (satisfies).

Concept

Simultaneous equations are two equations with two unknowns (e.g., x, y) solved together, often for quantities like tickets or items. Methods:

• Elimination: Add/subtract equations to eliminate a variable.
• Substitution: Express one variable in terms of the other and substitute.

UCAT keeps these simple, with integer solutions for efficiency.

Example 1: Elimination

2 adult and 3 child tickets cost 46; 1 adult and 2 child tickets cost 28. Find the cost of each.

Solution:

Let a = adult, c = child.

2a + 3c = 46 (1)

a + 2c = 28 (2)

Multiply (2) by 2: 2a + 4c = 56 (3)

Subtract (1) from (3): (2a + 4c) - (2a + 3c) = 56 - 46

c = 10

Substitute c = 10 in (2): a + 2 × 10 = 28

a + 20 = 28 → a = 8

Answer: Adult = 8, Child = 10

UCAT Tip: Elimination is faster here. Check: 2 × 8 + 3 × 10 = 16 + 30 = 46.

Example 2: Substitution

3x + y = 25; x + y = 15. Find x and y.

Solution:

From (2): y = 15 - x

Substitute into (1): 3x + (15 - x) = 25

3x + 15 - x = 25

2x = 25 - 15 = 10

x = 10 ÷ 2 = 5

Substitute x = 5: y = 15 - 5 = 10

Answer: x = 5, y = 10

UCAT Tip: Substitution works when one equation is simple. Check: 3 × 5 + 10 = 25.

Concept

Arithmetic sequences have a constant difference (d) between terms (e.g., 3, 7, 11, d = 4). The nth term is given by:

a + (n-1)d, where a is the first term.

UCAT uses sequences for cumulative costs, quantities, or time-based patterns. You may need to:

• Identify the pattern.
• Calculate a specific term.
• Sum terms (rare, but possible).

Example 1: Finding a Term

A clinic’s weekly cost increases: 40, 45, 50, 55, … What is the cost in week 8?

Solution:

First term (a) = 40, common difference (d) = 5

nth term: 40 + (n-1) × 5

Week 8 (n = 8): 40 + (8-1) × 5 = 40 + 7 × 5 = 40 + 35 = 75

Answer: 75

UCAT Tip: List terms: 40, 45, 50, 55, 60, 65, 70, 75. Use the formula for larger n.

Example 2: Identifying the Pattern

Costs are 10, 13, 16, 19, … Find the cost in week 5.

Solution:

a = 10, d = 3

nth term: 10 + (n-1) × 3

Week 5 (n = 5): 10 + (5-1) × 3 = 10 + 4 × 3 = 10 + 12 = 22

Answer: 22

UCAT Tip: Spot d quickly: 13 - 10 = 3. Check: 10, 13, 16, 19, 22.

Concept

Word problems require translating real-world scenarios into equations or formulas. Steps:

1. Identify unknowns and assign variables.
2. Write equations based on relationships.
3. Solve and interpret in context.

UCAT word problems involve costs, quantities, distances, or time, often with multiple constraints.

Example 1: Single Equation

A pharmacy buys 4 pens and 3 notebooks for 41, with each pen costing 5. Find the cost of a notebook.

Solution:

Let n = notebook cost.

4 × 5 + 3n = 41

20 + 3n = 41

3n = 41 - 20 = 21

n = 21 ÷ 3 = 7

Answer: 7

UCAT Tip: Extract numbers: 4 pens × 5 = 20. Check: 4 × 5 + 3 × 7 = 20 + 21 = 41.

Example 2: Multiple Constraints

A car travels 240 km. If it travels at 80 km/h for x hours and 60 km/h for y hours, and total time is 3.5 hours, find x.

Solution:

Distance: 80x + 60y = 240

Time: x + y = 3.5

From (2): y = 3.5 - x

Substitute: 80x + 60(3.5 - x) = 240

80x + 210 - 60x = 240

20x = 240 - 210 = 30

x = 30 ÷ 20 = 1.5

Answer: 1.5 hours

UCAT Tip: Use the calculator for decimals. Check: y = 3.5 - 1.5 = 2, 80 × 1.5 + 60 × 2 = 120 + 120 = 240.

Concept

Error checking ensures solutions are correct by substituting back into equations. Estimation simplifies calculations when answer options are far apart or time is limited. Techniques:

• Substitution: Plug solutions into the original equation.
• Rounding: Approximate coefficients or values for quick checks.
• Option Testing: Try answer choices in equations.

These are critical for UCAT’s multiple-choice format.

Example 1: Error Checking

Solve 4x + 10 = 34, then check.

Solution:

4x + 10 = 34

4x = 34 - 10 = 24

x = 24 ÷ 4 = 6

Check: 4 × 6 + 10 = 24 + 10 = 34

Answer: x = 6

UCAT Tip: Always substitute back in UCAT to catch errors, especially with decimals.

Example 2: Estimation

Solve 2.8x + 4.9 ≈ 18.5. Estimate if options are 4, 8, 12.

Solution:

Round: 3x + 5 ≈ 19

3x ≈ 19 - 5 = 14

x ≈ 14 ÷ 3 ≈ 4.67

Choose 4 (closest).

Precise: 2.8x = 18.5 - 4.9 = 13.6, x ≈ 4.86

Answer: ~4

UCAT Tip: Estimation saves time when options are spread out. Use the calculator for precise work if needed.

Concept

UCAT’s time constraint (40 seconds per question) requires strategic approaches to algebra:

• Mental Math vs. Calculator: Use mental math for simple equations (e.g., 2x = 12) to save time; reserve the calculator for complex substitutions or decimals.
• Test Answer Options: For multiple-choice questions, plug options into equations to find the correct one quickly.
• Simplify Early: Reduce expressions or equations before solving to minimize errors.
• Time Management: Flag and skip questions taking >40 seconds; return if time allows.
• Estimation: Use when answer options are far apart to eliminate wrong choices.
• Context Awareness: Ensure solutions make sense (e.g., positive quantities, correct units).

Tips for Success

• Practice solving equations mentally for speed (e.g., 3x = 15 → x = 5).
• Use the UCAT calculator for decimals or large numbers, but practice with the official tutorial to master its interface.
• Test options in equations like 5x + 10 = 35: try x = 5 (25 + 10 = 35, correct).
• Simplify expressions (e.g., 6x + 3x = 9x) before substituting values.
• Check units (e.g., hours vs. minutes) to avoid errors in formulas.
• If stuck, estimate or try a middle option to move forward.

Question 1: Linear Equation

A clinic’s cost is 5x + 15 = 45 for x kits. How many kits were bought?

Question 2: Formula

A car travels 200 km in 4 hours. Find speed: s = d/t.

Question 3: Simultaneous Equations

4 adult and 3 child tickets cost 68; 2 adult and 2 child tickets cost 40. Find the cost of an adult ticket.

Question 4: Word Problem

A pharmacy spends 3x + 2y = 50, where x is the cost of a pen and y is a notebook. If x = 10, find y.

Question 5: Sequence

Costs are 15, 20, 25, 30, … Find the cost in week 7.