USTET Mathematics — Ratio & ProportionExam Answer Templates

Marks

2

Topic

Basic Proportion

Difficulty

easy

Template Id

T1

Examiner Tip

Always show the cross-multiplication step even if you can solve mentally

Model Answer

Given: 3:5 = x:20 To Find: Value of x Solution: Using cross-multiplication: 3 × 20 = 5 × x 60 = 5x x = 60 ÷ 5 x = 12 Therefore, x = 12

Question Type

short_answer

Answer Structure

• Line 1: Write the given proportion [0.5 mark]
• Line 2: Apply cross-multiplication correctly [1 mark]
• Line 3: Solve the equation and box the answer [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• Not showing cross-multiplication step
• Arithmetic errors
• Not writing final answer clearly

Key Phrases To Include

• cross-multiplication
• Therefore

Marks

3

Topic

Partitive Proportion

Difficulty

medium

Template Id

T2

Examiner Tip

Always verify your answer by checking if the sum matches the given condition

Model Answer

Given: Ratio of two numbers = 3:4, Sum = 84 To Find: The two numbers Solution: Let the two numbers be 3x and 4x (where x is the common factor) Sum of the numbers = 3x + 4x = 7x According to the given condition: 7x = 84 x = 84 ÷ 7 = 12 Therefore: First number = 3x = 3 × 12 = 36 Second number = 4x = 4 × 12 = 48 Verification: 36 + 48 = 84 ✓ Therefore, the two numbers are 36 and 48.

Question Type

short_answer

Answer Structure

• Line 1-2: Define variables using the ratio [1 mark]
• Line 3-4: Set up equation using given condition [1 mark]
• Line 5-7: Solve and find both numbers [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not using ratio form (3x, 4x)
• Not verifying the answer
• Calculation errors

Key Phrases To Include

• Let the numbers be
• common factor
• Therefore
• Verification

Marks

3

Topic

Inverse Proportion

Difficulty

medium

Template Id

T3

Examiner Tip

Always identify whether the proportion is direct or inverse before solving

Model Answer

Given: 15 workers complete job in 12 days To Find: Number of workers needed to complete job in 9 days Solution: This is a case of inverse proportion (more workers → less time) Let the required number of workers = x Using inverse proportion: Workers × Days = Constant 15 × 12 = x × 9 180 = 9x x = 180 ÷ 9 = 20 Therefore, 20 workers are needed to complete the job in 9 days.

Question Type

short_answer

Answer Structure

• Line 1: Identify type of proportion (inverse) [1 mark]
• Line 2-3: Set up inverse proportion equation [1 mark]
• Line 4-5: Solve correctly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Treating as direct proportion
• Not stating the type of proportion
• Wrong equation setup

Key Phrases To Include

• inverse proportion
• more workers → less time
• constant

Marks

3

Topic

Partitive Proportion

Difficulty

medium

Template Id

T4

Examiner Tip

Always verify that the individual shares add up to the original total

Model Answer

Given: Total amount = ₱1200, Ratio A:B:C = 2:3:7 To Find: Share of each person Solution: Sum of ratio parts = 2 + 3 + 7 = 12 Value of one part = ₱1200 ÷ 12 = ₱100 Share of A = 2 × ₱100 = ₱200 Share of B = 3 × ₱100 = ₱300 Share of C = 7 × ₱100 = ₱700 Verification: ₱200 + ₱300 + ₱700 = ₱1200 ✓ Therefore, A gets ₱200, B gets ₱300, and C gets ₱700.

Question Type

short_answer

Answer Structure

• Line 1: Find sum of ratio parts [1 mark]
• Line 2: Calculate value of one part [1 mark]
• Line 3-5: Find individual shares and verify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not finding sum of parts first
• Calculation errors
• Not including units (₱)

Key Phrases To Include

• sum of ratio parts
• value of one part
• Verification

Marks

5

Topic

Partitive Proportion

Difficulty

medium

Template Id

T5

Examiner Tip

For 5-mark questions, always show detailed step-by-step solution with proper verification

Model Answer

Given: Ratio of boys to girls = 4:5, Total students = 180 To Find: Number of boys and girls Solution: Step 1: Understanding the ratio Boys : Girls = 4 : 5 This means for every 4 boys, there are 5 girls. Step 2: Find the sum of ratio parts Sum of ratio parts = 4 + 5 = 9 parts Step 3: Find the value of one part Total students = 180 Value of one part = 180 ÷ 9 = 20 students Step 4: Calculate number of boys and girls Number of boys = 4 parts = 4 × 20 = 80 boys Number of girls = 5 parts = 5 × 20 = 100 girls Step 5: Verification Total = 80 + 100 = 180 ✓ Ratio check: 80:100 = 4:5 (dividing by 20) ✓ Therefore, there are 80 boys and 100 girls in the school.

Question Type

long_answer

Answer Structure

• Step 1: Interpret the given ratio [1 mark]
• Step 2: Find sum of ratio parts [1 mark]
• Step 3: Calculate value of one part [1 mark]
• Step 4: Find number of boys and girls [1 mark]
• Step 5: Verify the answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not showing step-by-step solution
• Missing verification
• Calculation errors
• Not labeling clearly

Key Phrases To Include

• sum of ratio parts
• value of one part
• verification
• ratio check

Marks

2

Topic

Ratio Simplification

Difficulty

easy

Template Id

T6

Examiner Tip

Always check if the ratio can be simplified further by finding the HCF

Model Answer

Given: 0.6 To Find: Ratio in simplest form Solution: 0.6 = 6/10 Expressing as ratio: 6:10 Simplifying by dividing both terms by their HCF: HCF of 6 and 10 = 2 6 ÷ 2 : 10 ÷ 2 = 3:5 Therefore, 0.6 = 3:5 in simplest form.

Question Type

short_answer

Answer Structure

• Line 1: Convert decimal to fraction [0.5 mark]
• Line 2: Express as ratio and find HCF [1 mark]
• Line 3: Simplify to lowest terms [0.5 mark]

Scoring Breakdown

Common Mark Deductions

• Not simplifying to lowest terms
• Calculation errors
• Not showing HCF

Key Phrases To Include

• HCF
• simplest form
• dividing both terms

Marks

3

Topic

Compound Ratios

Difficulty

hard

Template Id

T7

Examiner Tip

When combining ratios, always equalize the common term first

Model Answer

Given: a:b = 2:3 and b:c = 4:5 To Find: a:b:c Solution: From a:b = 2:3, we get a/b = 2/3 From b:c = 4:5, we get b/c = 4/5 To combine ratios, make the value of b same in both ratios: a:b = 2:3 = 2×4 : 3×4 = 8:12 b:c = 4:5 = 4×3 : 5×3 = 12:15 Since b = 12 in both ratios: a:b:c = 8:12:15 Therefore, a:b:c = 8:12:15

Question Type

short_answer

Answer Structure

• Line 1: Write both given ratios clearly [0.5 mark]
• Line 2-4: Make b equal in both ratios [1.5 marks]
• Line 5: Combine to get final ratio [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not equalizing b values
• Calculation errors
• Wrong final ratio

Key Phrases To Include

• make the value of b same
• combine ratios

Marks

3

Topic

Direct Proportion Applications

Difficulty

medium

Template Id

T8

Examiner Tip

For speed-distance-time problems, identify what remains constant to determine proportion type

Model Answer

Given: Speed ratio = 3:4, First car travels 180 km To Find: Distance traveled by second car in same time Solution: This is direct proportion (same time, speed ∝ distance) Let speed of first car = 3x and speed of second car = 4x Distance traveled by first car = 180 km Using the relationship: Distance = Speed × Time For same time: Distance ∝ Speed Therefore: Distance₁/Distance₂ = Speed₁/Speed₂ 180/Distance₂ = 3/4 Distance₂ = (180 × 4)/3 = 720/3 = 240 km Therefore, the second car will travel 240 km.

Question Type

short_answer

Answer Structure

• Line 1: Identify as direct proportion [1 mark]
• Line 2-3: Set up proportion equation [1 mark]
• Line 4-5: Solve correctly [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not identifying proportion type
• Wrong equation setup
• Calculation errors

Key Phrases To Include

• direct proportion
• same time
• Distance ∝ Speed

Marks

2

Topic

Types of Proportion

Difficulty

easy

Template Id

T9

Examiner Tip

Always provide reasoning for your identification of proportion type

Model Answer

Solution: (a) Number of workers and time to complete a job: Inverse proportion Reason: More workers → Less time needed As one quantity increases, the other decreases. (b) Speed of a car and time to cover a fixed distance: Inverse proportion Reason: Higher speed → Less time needed to cover same distance As one quantity increases, the other decreases. Therefore: (a) Inverse proportion (b) Inverse proportion

Question Type

short_answer

Answer Structure

• Line 1: Identify type for part (a) with reason [1 mark]
• Line 2: Identify type for part (b) with reason [1 mark]

Scoring Breakdown

Common Mark Deductions

• No reasoning provided
• Confusing direct and inverse
• Incomplete answers

Key Phrases To Include

• More workers → Less time
• Higher speed → Less time
• inverse proportion

Marks

3

Topic

Scale and Maps

Difficulty

medium

Template Id

T10

Examiner Tip

Pay attention to units - scale problems often require unit conversion

Model Answer

Given: Map scale = 1:50000, Map distance = 4 cm To Find: Actual distance in km Solution: Scale 1:50000 means 1 cm on map = 50000 cm in reality Map distance = 4 cm Actual distance = 4 × 50000 = 200000 cm Converting to km: 200000 cm = 200000 ÷ 100000 km = 2 km (Since 1 km = 100000 cm) Therefore, the actual distance between the cities is 2 km.

Question Type

short_answer

Answer Structure

• Line 1: Interpret the scale correctly [1 mark]
• Line 2: Calculate actual distance in cm [1 mark]
• Line 3: Convert to km [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not interpreting scale correctly
• Forgetting unit conversion
• Calculation errors

Key Phrases To Include

• scale means
• converting to km
• 1 km = 100000 cm

Marks

3

Topic

Algebraic Ratios

Difficulty

hard

Template Id

T11

Examiner Tip

Always verify your answer by substituting back into the original ratio

Model Answer

Given: (x+1):(x+3) = 3:5 To Find: Value of x Solution: Writing as fractions: (x+1)/(x+3) = 3/5 Cross-multiplying: 5(x+1) = 3(x+3) 5x + 5 = 3x + 9 5x - 3x = 9 - 5 2x = 4 x = 2 Verification: When x = 2 (x+1):(x+3) = (2+1):(2+3) = 3:5 ✓ Therefore, x = 2

Question Type

short_answer

Answer Structure

• Line 1: Set up equation using cross-multiplication [1 mark]
• Line 2-4: Solve the linear equation [1 mark]
• Line 5: Verify the answer [1 mark]

Scoring Breakdown

Common Mark Deductions

• Algebraic errors
• Not expanding brackets correctly
• No verification

Key Phrases To Include

• cross-multiplying
• linear equation
• verification

Marks

5

Topic

Age Problems with Ratios

Difficulty

hard

Template Id

T12

Examiner Tip

For age problems, always verify both present and future ratios

Model Answer

Given: Present age ratio A:B = 4:3, After 5 years ratio = 9:7 To Find: Present ages of A and B Solution: Step 1: Let present ages be in ratio 4:3 Let A's present age = 4x years Let B's present age = 3x years Step 2: Set up equation for ages after 5 years After 5 years: A's age = (4x + 5) years B's age = (3x + 5) years Ratio after 5 years = (4x + 5):(3x + 5) = 9:7 Step 3: Solve using cross-multiplication (4x + 5)/(3x + 5) = 9/7 7(4x + 5) = 9(3x + 5) 28x + 35 = 27x + 45 28x - 27x = 45 - 35 x = 10 Step 4: Find present ages A's present age = 4x = 4 × 10 = 40 years B's present age = 3x = 3 × 10 = 30 years Step 5: Verification Present ratio: 40:30 = 4:3 ✓ After 5 years: A = 45, B = 35 Ratio after 5 years: 45:35 = 9:7 ✓ Therefore, A's present age is 40 years and B's present age is 30 years.

Question Type

long_answer

Answer Structure

• Step 1: Define variables using given ratio [1 mark]
• Step 2: Express ages after 5 years [1 mark]
• Step 3: Set up and solve equation [2 marks]
• Step 4: Find actual ages and verify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not defining variables clearly
• Algebraic errors
• Missing verification
• Wrong final ages

Key Phrases To Include

• Let present ages
• After 5 years
• cross-multiplication
• verification

Marks

1

Topic

Decimal Ratios

Difficulty

easy

Template Id

T13

Examiner Tip

Convert decimals to whole numbers before simplifying ratios

Model Answer

2.4:3.6 = 24:36 = 2:3

Question Type

very_short_answer

Answer Structure

• Convert to whole numbers and simplify [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not simplifying
• Keeping in decimal form

Key Phrases To Include

• simplifying
• whole numbers

Marks

2

Topic

Unitary Method

Difficulty

easy

Template Id

T14

Examiner Tip

Include currency units in your final answer

Model Answer

Given: 3 kg costs ₱150 To Find: Cost of 7 kg Solution: Using direct proportion (more quantity → more cost): 3 kg : ₱150 = 7 kg : x 3/150 = 7/x 3x = 7 × 150 = 1050 x = 1050 ÷ 3 = ₱350 Therefore, 7 kg will cost ₱350.

Question Type

short_answer

Answer Structure

• Line 1: Set up proportion [1 mark]
• Line 2: Solve using cross-multiplication [1 mark]

Scoring Breakdown

Common Mark Deductions

• Wrong proportion setup
• Calculation errors
• Missing units

Key Phrases To Include

• direct proportion
• cross-multiplication

Marks

3

Topic

Geometry with Ratios

Difficulty

medium

Template Id

T15

Examiner Tip

Always include degree symbols and verify that angles sum to 180°

Model Answer

Given: Angles are in ratio 2:3:4 To Find: All three angles Solution: Let the angles be 2x, 3x, and 4x Sum of angles in a triangle = 180° 2x + 3x + 4x = 180° 9x = 180° x = 20° Therefore: First angle = 2x = 2 × 20° = 40° Second angle = 3x = 3 × 20° = 60° Third angle = 4x = 4 × 20° = 80° Verification: 40° + 60° + 80° = 180° ✓

Question Type

short_answer

Answer Structure

• Line 1: Express angles using ratio (2x, 3x, 4x) [1 mark]
• Line 2: Use angle sum property to find x [1 mark]
• Line 3: Calculate all three angles [1 mark]

Scoring Breakdown

Common Mark Deductions

• Not using angle sum property
• Calculation errors
• Missing degree symbols

Key Phrases To Include

• sum of angles
• triangle
• verification