Numbers, Quantification and Numerical Applications

The One Thing

Numerical applications require the accurate translation of quantities and relationships into mathematical form. Reliable solutions depend on correct operations, compatible units, appropriate precision, dimensional consistency and interpretation of the final result in context.

Definitions and Results

• Natural Numbers: Counting numbers such as $1,2,3,\dots$; some conventions also include $0$.
• Whole Numbers: Natural numbers together with zero: $0,1,2,3,\dots$.
• Integers: Positive and negative whole numbers, including zero.
• Rational Numbers: Numbers expressible as $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.
• Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers, such as $\sqrt{2}$ and $\pi$.
• Real Numbers: All rational and irrational numbers represented on the number line.
• Absolute Value: The distance of a number from zero, written $|x|$; it is always non-negative.
• Ratio: A comparison of two quantities expressed in the same units, such as $a:b$.
• Proportion: An equality between two ratios, such as $\frac{a}{b}=\frac{c}{d}$.
• Percentage: A ratio expressed out of $100$; $x\%=\frac{x}{100}$.
• Average or Arithmetic Mean: The sum of all observations divided by the number of observations.
• Weighted Mean: An average in which observations have different levels of importance.
• Unitary Method: A method that first determines the value of one unit and then uses it to calculate the required quantity.
• Approximation: A value close to the exact value, used when exact calculation is unnecessary or difficult.
• Significant Figures: Digits that convey meaningful precision in a measured or calculated value.
• Scientific Notation: A number written as $a\times 10^n$, where $1\le |a|<10$ and $n$ is an integer.
• Sequence: An ordered list of numbers following a stated pattern or rule.
• Arithmetic Progression: A sequence in which the difference between consecutive terms is constant.
• Geometric Progression: A sequence in which the ratio between consecutive non-zero terms is constant.
• Factorial: For a non-negative integer $n$, $n!=n(n-1)(n-2)\cdots 2\times 1$, with $0!=1$.
• Permutation: An arrangement of objects in which order matters.
• Combination: A selection of objects in which order does not matter.
• Numerical Reasoning: The ability to interpret numerical information and apply mathematical operations logically.
• Number-system inclusion: Natural numbers are contained in whole numbers, whole numbers in integers, integers in rational numbers, and rational numbers in real numbers.
• Percentage change: $\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100$.
• Simple Interest: $\mathrm{SI}=\frac{PRT}{100}$, where $P$ is principal, $R$ is the annual rate in percent and $T$ is time in years.
• Amount under Simple Interest: $A=P+\mathrm{SI}$.
• Compound Amount: $A=P(1+R/100{)}^n$ for annual compounding, where $n$ is the number of compounding periods.
• Compound Interest: $\mathrm{CI}=A-P$.
• Speed, Distance and Time: $\text{speed}=\frac{\text{distance}}{\text{time}}$, $\text{distance}=\text{speed}\times \text{time}$, and $\text{time}=\frac{\text{distance}}{\text{speed}}$.
• Average speed for equal distances: $\frac{2uv}{u+v}$, where $u$ and $v$ are the two speeds.
• Arithmetic Progression: $a_n=a+(n-1)d$ and $S_n=\frac{n}{2}[2a+(n-1)d]$.
• Geometric Progression: $a_n=a{r}^{n-1}$ and, for $r\ne 1$, $S_n=\frac{a(r^n-1)}{r-1}$.
• Permutation formula: ${}^n{P}_r=\frac{n!}{(n-r)!}$.
• Combination formula: ${}^n{C}_r=\frac{n!}{r!(n-r)!}$.
• Permutation–combination relationship: ${}^n{P}_r={}^n{C}_r\times r!$, because each selection can be arranged in $r!$ orders.
• Dimensional consistency: Both sides of a valid equation must have the same physical units.

Worked Methods

1. Translating quantities into mathematical form

• Identify the known quantities.
• Identify the unknown quantity.
• Establish the relevant relationship, formula or proportion.
• Convert units where necessary.
• Perform the calculation.
• Interpret the result in the language of the real situation.

Equivalent forms—fractions, decimals, percentages and ratios—should be selected according to context. Ratio comparisons require compatible units before simplification.

2. Percentage increase and decrease

For an increase of $r\%$:

$$\text{new value}=\text{original value}\left(1+\frac{r}{100}\right).$$

For a decrease of $r\%$:

$$\text{new value}=\text{original value}\left(1-\frac{r}{100}\right).$$

The percentage change is:

$$\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100.$$

The sign of the result indicates whether the quantity has increased or decreased.

3. Simple and compound interest

For simple interest:

• Identify $P$, $R$ and $T$.
• Substitute into

$$\mathrm{SI}=\frac{PRT}{100}.$$

• Find the total amount using

$$A=P+\mathrm{SI}.$$

For compound interest with annual compounding:

• Identify the principal $P$, annual rate $R$, and number of compounding periods $n$.
• Calculate

$$A=P(1+R/100{)}^n.$$

• Calculate compound interest:

$$\mathrm{CI}=A-P.$$

4. Speed, distance and time

• Identify which of distance, speed or time is unknown.
• Use the appropriate relationship:

$$\text{speed}=\frac{\text{distance}}{\text{time}},$$ $$\text{distance}=\text{speed}\times \text{time},$$ $$\text{time}=\frac{\text{distance}}{\text{speed}}.$$

• Convert units consistently.
• Check that the final unit matches the quantity required.

For equal distances travelled at speeds $u$ and $v$, do not use the ordinary arithmetic mean of the speeds. Use:

$$\text{average speed}=\frac{2uv}{u+v}.$$

5. Arithmetic progressions

Given first term $a$, common difference $d$, and term number $n$:

• Use the nth-term formula:

$$a_n=a+(n-1)d.$$

• For the sum of the first $n$ terms, use:

$$S_n=\frac{n}{2}[2a+(n-1)d].$$

• Substitute the values while preserving the sign of $d$.

6. Geometric progressions

Given first term $a$, common ratio $r$, and term number $n$:

• Use the nth-term formula:

$$a_n=a{r}^{n-1}.$$

• For $r\ne 1$, use:

$$S_n=\frac{a(r^n-1)}{r-1}.$$

• Ensure that the ratio is taken between consecutive non-zero terms.

7. Permutations and combinations

For arrangements in which order matters:

• Identify $n$, the total number of objects, and $r$, the number selected.
• Use:

$${}^n{P}_r=\frac{n!}{(n-r)!}.$$

For selections in which order does not matter:

• Identify $n$ and $r$.
• Use:

$${}^n{C}_r=\frac{n!}{r!(n-r)!}.$$

The relationship

$${}^n{P}_r={}^n{C}_r\times r!$$

shows that arrangements equal selections multiplied by the number of possible orders.

8. Estimation, approximation and checking

• Determine the required degree of accuracy.
• Approximate only when exact calculation is unnecessary or difficult.
• Round at the final step whenever possible to reduce accumulated error.
• Check the result for reasonable units, sign, size and context.
• Where possible, verify it by estimation, an alternative method or dimensional consistency.

A negative answer may represent direction, loss, decrease or debt, depending on the situation. Dates, time intervals, money, distance, mass and temperature must be handled using their appropriate units and conversion rules.

Where It Goes Wrong

• Treating rational numbers as allowing any denominator; the denominator must never be zero.
• Comparing or simplifying ratios before converting quantities into compatible units.
• Applying percentage change without using the original value as the denominator.
• Confusing simple interest with compound interest, or omitting the conditions represented by $T$, $n$ and annual compounding.
• Using the ordinary mean instead of $\frac{2uv}{u+v}$ for equal distances travelled at speeds $u$ and $v$.
• Rounding too early, failing to check dimensional consistency, or rejecting a negative result without considering whether it represents direction, loss, decrease or debt.

What Gets Asked

• Classify numbers as natural, whole, integer, rational, irrational or real, and state the inclusion relationships.
• Calculate absolute values and distinguish equivalent fractions, decimals, percentages and ratios.
• Simplify ratios and solve proportions after converting units where necessary.
• Find arithmetic means, weighted means and values using the unitary method.
• Calculate percentage change and determine new values after percentage increases or decreases.
• Solve simple-interest and compound-interest problems, including amounts and compound interest.
• Use speed–distance–time relationships and calculate average speed for equal distances.
• Determine terms and sums in arithmetic and geometric progressions.
• Evaluate factorials and distinguish permutations from combinations.
• Express numbers in scientific notation and apply the laws of exponents:

$$a^m\times a^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n}(a\ne 0),(a^m{)}^n=a^{mn}.$$

• Round values to an appropriate number of significant figures and judge the reasonableness of estimates.
• Check numerical answers through units, signs, magnitude, alternative methods and the context of dates, time intervals, money, distance, mass or temperature.