Civil Service Exam (Professional) Numerical Ability — Decimals & Scientific NotationExam Answer Templates

Examiner tip

Count decimal places moved to determine the exponent - moving left gives positive exponent, moving right gives negative

Model answer

0.00356 = 3.56 × 10^(-3)

Answer structure

• Write the number in a × 10^n format where 1 ≤ a < 10 [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect exponent sign
• Coefficient not between 1 and 10
• Missing multiplication sign

Key phrases to include

• × 10^
• coefficient between 1 and 10

Examiner tip

Always align decimal points and add trailing zeros to make all numbers have the same number of decimal places

Model answer

3.450
0.678
+12.900

17.028

Answer structure

• Align decimal points vertically [1 mark]
• Add correctly and show final answer [1 mark]

Scoring breakdown

Common mark deductions

• Misaligned decimal points
• Calculation errors
• Not showing working

Key phrases to include

• align decimal points
• add zeros for equal decimal places

Examiner tip

When squaring scientific notation, square the coefficient and double the exponent

Model answer

Step 1: Convert to standard form
4.56 × 10^5 = 456,000

Step 2: Calculate the square
(456,000)² = (4.56 × 10^5)²
= (4.56)² × (10^5)²
= 20.7936 × 10^10
= 2.07936 × 10^11

Answer structure

• Convert scientific notation to standard form [1 mark]
• Set up the square calculation [1 mark]
• Calculate and express answer in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect decimal place movement
• Wrong exponent calculation
• Coefficient not between 1 and 10

Key phrases to include

• standard form
• square both coefficient and power
• proper scientific notation

Examiner tip

Make divisor a whole number by multiplying both dividend and divisor by the same power of 10

Model answer

2.88 ÷ 0.036
= 2880 ÷ 36 (multiply both by 1000)
= 80

Therefore, 2.88 ÷ 0.036 = 80

Answer structure

• Convert division to whole numbers by multiplying both by appropriate power of 10 [1 mark]
• Perform division and state final answer [1 mark]

Scoring breakdown

Common mark deductions

• Not showing conversion step
• Calculation errors
• Incorrect decimal elimination

Key phrases to include

• multiply both by same power of 10
• eliminate decimals

Examiner tip

Always start with given information and show each step clearly for maximum marks

Model answer

Given:

• Initial bacteria count = 2.5 × 10^4
• Doubling time = 3 hours
• Total time = 12 hours

Step 1: Find number of doubling periods
Number of periods = 12 ÷ 3 = 4 periods

Step 2: Calculate growth factor
After 4 doubling periods, bacteria multiply by 2^4 = 16

Step 3: Calculate final bacteria count
Final count = Initial count × Growth factor
= 2.5 × 10^4 × 16
= 2.5 × 16 × 10^4
= 40 × 10^4
= 4.0 × 10^5

Therefore, after 12 hours, there will be 4.0 × 10^5 bacteria.

Answer structure

• Write given information clearly [1 mark]
• Calculate number of doubling periods [1 mark]
• Determine growth factor (2^4 = 16) [1 mark]
• Calculate final bacteria count [1 mark]
• Express answer in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Not stating given information
• Incorrect period calculation
• Wrong final notation format
• Calculation errors

Key phrases to include

• given information
• doubling periods
• growth factor
• scientific notation

Examiner tip

Look at the third decimal place - if 5 or more, round up

Model answer

23.4567 rounded to 2 decimal places = 23.46

Answer structure

• Apply rounding rule and give correct answer [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect rounding
• Wrong number of decimal places

Key phrases to include

• rounded to
• decimal places

Examiner tip

When multiplying scientific notation, multiply coefficients and add exponents

Model answer

(3.2 × 10^6) × (1.5 × 10^(-3))
= 3.2 × 1.5 × 10^6 × 10^(-3)
= 4.8 × 10^(6-3)
= 4.8 × 10^3

Answer structure

• Multiply coefficients and add exponents [1 mark]
• Express answer in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Subtracting instead of adding exponents
• Incorrect coefficient multiplication

Key phrases to include

• multiply coefficients
• add exponents

Examiner tip

Count how many places the decimal moves from its original position

Model answer

456,000,000 = 4.56 × 10^8

Answer structure

• Write in a × 10^n format where 1 ≤ a < 10 [1 mark]

Scoring breakdown

Common mark deductions

• Wrong exponent
• Coefficient outside 1-10 range

Key phrases to include

• × 10^
• move decimal point

Examiner tip

Add zeros to make all numbers have the same decimal places for easier calculation

Model answer

15.600

• 8.970

+3.045

9.675

Answer structure

• Align decimal points and add trailing zeros [1 mark]
• Perform calculations correctly [1 mark]

Scoring breakdown

Common mark deductions

• Misalignment
• Calculation errors
• Not showing working

Key phrases to include

• align decimal points
• trailing zeros

Examiner tip

For reciprocal of scientific notation, find reciprocal of coefficient and change sign of exponent

Model answer

Step 1: Convert to scientific notation
0.000034 = 3.4 × 10^(-5)

Step 2: Find reciprocal
Reciprocal = 1/(3.4 × 10^(-5))
= 1/3.4 × 1/10^(-5)
= 0.294 × 10^5
= 2.94 × 10^4

Answer structure

• Convert decimal to scientific notation [1 mark]
• Set up reciprocal calculation [1 mark]
• Calculate reciprocal and express in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect initial conversion
• Wrong reciprocal calculation
• Improper final notation

Key phrases to include

• scientific notation
• reciprocal
• invert coefficient and exponent

Examiner tip

Always start by clearly stating what is given and what needs to be found

Model answer

Given:

• 1 light-year = 9.46 × 10^12 km
• Distance = 4.2 light-years

Distance in km = 4.2 × 9.46 × 10^12
= 39.732 × 10^12
= 3.9732 × 10^13 km

Therefore, the nearest star is 3.97 × 10^13 km away.

Answer structure

• Write given information [1 mark]
• Set up multiplication correctly [1 mark]
• Calculate and express in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Not stating given information
• Calculation errors
• Incorrect final notation

Key phrases to include

• given
• light-year
• scientific notation

Examiner tip

When dividing scientific notation, divide coefficients and subtract exponents

Model answer

(2.4 × 10^3) ÷ (6.0 × 10^(-2))
= (2.4 ÷ 6.0) × 10^(3-(-2))
= 0.4 × 10^5
= 4.0 × 10^4

Answer structure

• Divide coefficients and subtract exponents [1 mark]
• Express final answer in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Adding instead of subtracting exponents
• Incorrect coefficient division

Key phrases to include

• divide coefficients
• subtract exponents

Examiner tip

Always state the formulas before substituting values

Model answer

Given:
Length = 45.6 m
Width = 23.75 m

Area = Length × Width
= 45.6 × 23.75
= 1083.0 m²

Perimeter = 2(Length + Width)
= 2(45.6 + 23.75)
= 2(69.35)
= 138.7 m

Answer structure

• Write given information and formulas [1 mark]
• Calculate area correctly [1 mark]
• Calculate perimeter correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not stating formulas
• Calculation errors
• Missing units

Key phrases to include

• given
• area formula
• perimeter formula

Examiner tip

Add trailing zeros to compare decimals with different decimal places

Model answer

Comparing the decimals:
3.45 = 3.450
3.405 < 3.45 = 3.450 < 3.54

Ascending order: 3.405, 3.45, 3.450, 3.54
Note: 3.45 = 3.450

Answer structure

• Compare decimals by examining each decimal place [1 mark]
• Arrange in correct ascending order [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect comparison
• Wrong ordering

Key phrases to include

• compare
• ascending order
• decimal places

Examiner tip

This is a classic chemistry application - always relate to real-world significance like atomic mass

Model answer

Given:

• Mass of one hydrogen atom = 1.67 × 10^(-27) kg
• Number of atoms = 6.02 × 10^23 (Avogadro's number)

To find: Total mass of 6.02 × 10^23 hydrogen atoms

Solution:
Total mass = Number of atoms × Mass per atom
= (6.02 × 10^23) × (1.67 × 10^(-27))
= 6.02 × 1.67 × 10^23 × 10^(-27)
= 10.0534 × 10^(23-27)
= 10.0534 × 10^(-4)
= 1.00534 × 10^(-3) kg
≈ 1.01 × 10^(-3) kg

Therefore, the mass of 6.02 × 10^23 hydrogen atoms is approximately 1.01 × 10^(-3) kg or 1.01 g.

Answer structure

• Write given information clearly [1 mark]
• State what needs to be found [1 mark]
• Set up the multiplication correctly [1 mark]
• Perform calculation with proper exponent handling [1 mark]
• Express final answer in proper scientific notation with appropriate rounding [1 mark]

Scoring breakdown

Common mark deductions

• Not stating given data
• Incorrect exponent calculation
• Wrong final notation
• No rounding indication

Key phrases to include

• given
• to find
• Avogadro's number
• scientific notation
• mass calculation

Do

• Give precise final answer
• Use correct notation
• Show key conversion step if needed

Don't

• Waste time on extensive working
• Give intermediate steps
• Leave answer unlabeled

Do

• Show alignment for decimal operations
• Display one calculation step
• Label final answer clearly

Don't

• Skip showing working
• Make calculation errors
• Forget to check decimal points

Do

• Number or label each step
• Show all significant calculations
• Verify scientific notation format

Don't

• Rush through steps
• Combine too many operations
• Forget units or proper notation

Do

• Start with given information
• Show complete working
• State final answer in context
• Use proper scientific notation throughout

Don't

• Skip any major step
• Make arithmetic errors
• Forget to relate answer to original question
• Use improper notation