UPCAT Mathematics — Ratio & ProportionExam Answer Templates

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

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

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.

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

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.

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

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.

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

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.

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

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.

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

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

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

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.

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

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

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

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.

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

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

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

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.

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

Examiner tip

Convert decimals to whole numbers before simplifying ratios

Model answer

2.4:3.6 = 24:36 = 2:3

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

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.

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

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° ✓

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

Do

• Write final answer clearly
• Show key calculation step
• Include units if required

Don't

• Don't over-explain
• Don't skip the calculation step
• Don't forget to simplify

Do

• Write Given and To Find
• Show cross-multiplication step
• Box final answer
• Include verification if time permits

Don't

• Don't skip intermediate steps
• Don't make arithmetic errors
• Don't forget units

Do

• Organize in clear steps
• Identify direct/inverse proportion
• Show all calculations
• Verify answer
• Use proper mathematical language

Don't

• Don't rush calculations
• Don't skip proportion identification
• Don't omit verification

Do

• Use Step 1, Step 2 format
• Explain your reasoning
• Show multiple verification checks
• Use proper mathematical terminology
• Include diagram if helpful

Don't

• Don't rush any step
• Don't skip explanations
• Don't omit any verification
• Don't make the solution too brief