Unit 6: Numerical Ability - Subjective Questions

Order of operations:

• Evaluate expressions inside parentheses first.
• Evaluate exponents or powers.
• Perform multiplication and division from left to right.
• Perform addition and subtraction from left to right.

For the given expression:

First, simplify the parentheses:

Therefore:

Perform division and multiplication from left to right:

Finally:

Answer: .

Relationship:

• A fraction represents a part of a whole as a ratio.
• A decimal expresses the same value using powers of .
• A percentage expresses the value as a part out of .

To convert the fraction into a decimal:

To convert the decimal into a percentage, multiply by :

Thus:

• Fraction:
• Decimal:
• Percentage:

The discount amount is:

The price after discount is:

Tax is calculated on the discounted price:

Therefore, the final amount is:

Why tax is not calculated on the marked price:

• The discount first reduces the selling price.
• The taxable value is therefore , not .
• Calculating tax directly on the marked price would incorrectly ignore the discount.

Final amount paid: .

Let the principal be , the annual interest rate be , and the number of years be .

After one year:

After two years:

Continuing this process for years gives:

For , , and :

Compound interest is:

• Amount:
• Compound interest:

The sum of the ratio terms is:

The value of one ratio unit is:

Therefore, the three parts are:

Verification:

Also:

Hence, the required parts are , , and .

Numerical estimation is the process of finding an approximate value by replacing numbers with nearby values that are easier to calculate.

Use the rounded values:

The estimated value is:

The actual value is:

The absolute difference is:

Thus:

• Estimated value:
• Actual value: approximately
• Absolute estimation error: approximately

The estimate is reasonably close and is useful for a quick check.

Rounding: The next digit is examined to decide whether the retained final digit should increase.

To three decimal places:

The fourth decimal digit is , so the third decimal digit is unchanged.

Truncation: Digits beyond the required position are simply removed without adjustment.

To three decimal places:

Significant figures: Digits are counted beginning with the first nonzero digit.

To three significant figures:

The first three significant digits are , , and . The next digit is , so rounds upward to . The zero indicates that the result has three significant figures.

Rounding and truncation happen to produce the same three-decimal-place value here, but they do not always give the same result.

Rounding to the nearest thousand gives:

Numbers from up to, but not including, round to .

Therefore, if is the actual population:

The maximum rounding error is half of :

Role of error bounds:

• Error bounds describe the interval in which the actual value must lie.
• They show the precision of the rounded figure.
• They prevent a rounded estimate from being treated as an exact value.

Rounded population: .

Possible range: .

Since:

it follows that:

Take the initial approximation as:

The Babylonian method is:

For :

Therefore:

The estimate is reasonable because is approximately .

Estimation checks whether a computed answer has the correct approximate size. It can reveal misplaced decimal points, extra zeros, or incorrect operations.

Round the factors:

The estimated product is:

Therefore, the exact answer should be close to , not approximately .

The exact product is:

The student's answer has two extra zeros.

Conclusion: The claim is unreasonable. The correct answer is .

Inductive numerical reasoning identifies a pattern from specific examples and proposes a general conclusion. The conclusion is probable but may require proof.

Example:

Each term appears to be twice the previous term, so one predicts that the next term is:

Deductive numerical reasoning applies a known rule or general principle to reach a logically certain conclusion.

Example: Every multiple of is even. Since:

is a multiple of , and therefore must be even.

Main distinction:

• Induction moves from examples to a general rule.
• Deduction moves from a general rule to a specific conclusion.

Examine the differences between consecutive terms:

The differences are consecutive even numbers:

The next differences are and . Therefore:

The terms can also be written as:

Hence, the th term is:

Verification for :

Thus, the next two terms are and , and the general term is .

Let the son's present age be years.

Then the father's present age is:

After years:

• Son's age will be .
• Father's age will be .

According to the condition:

Expand the right-hand side:

Therefore:

Thus, the son's present age is years, and the father's present age is:

Verification: After years, their ages will be and , and .

Answer: Son: years; father: years.

Average speed is defined as:

Time for the first part:

Time for the second part:

Total distance:

Total time:

Therefore:

The arithmetic mean of the speeds is also km/h in this particular case because the train spends equal time at both speeds. In general, speeds cannot simply be averaged unless the time intervals are equal. The reliable method is always total distance divided by total time.

Average speed: km/h.

Production is directly proportional to both the number of machines and the operating time.

The production rate per machine per hour is:

Thus, one machine produces components per hour.

Five machines working for hours produce:

Alternatively, using direct proportion:

• Increasing the machines from to multiplies production by .
• Increasing the time from to hours multiplies production by .

Answer: The five machines will produce components.

Total sales:

Monthly average:

Range:

The maximum value is , and the minimum value is .

Percentage increase from January to April:

The increase is:

Therefore:

Results:

• Total books sold:
• Average per month:
• Range:
• Percentage increase:

The number preferring the first three subjects is:

Therefore, the number preferring history is:

The percentage preferring history is:

The sector angle in a pie chart is:

A complete pie chart would use the same formula for every category:

History results:

• Number of students:
• Percentage:
• Pie-chart angle:

The data are already arranged in ascending order:

Mean:

Median: There are seven values, so the median is the fourth value:

Mode: The most frequently occurring value is:

Range:

Interpretation:

• The mean gives the arithmetic average score.
• The median gives the central score when the data are ordered.
• The mode identifies the most common score.
• The range measures the spread between the highest and lowest scores.

Therefore, the mean is , the median is , the mode is , and the range is .

The total annual revenue is:

The percentage contribution of each quarter is:

The strongest quarter is with lakhs, while the weakest is with lakhs.

The absolute growth from to is:

The percentage growth is:

Summary:

• :
• :
• :
• :
• First-to-fourth-quarter growth: lakhs or

Arrange the values in ascending order:

Mean:

Median: The middle value is:

The value is much larger than the remaining values, which are clustered between and . It is therefore a possible outlier.

Effect of the outlier:

• It pulls the mean upward from the low-eighties to .
• It does not greatly affect the median because the median depends on the middle position.
• The mean of does not resemble most of the daily counts.

The median of better represents a typical day for this data set. Before discarding , however, its cause should be investigated because it may represent a genuine event rather than an error.