6.1 Quantity, Number & Operations
Key Takeaways
- NBT Quantitative Literacy (QL) does not allow a calculator; every comparison and arithmetic step must be done by hand with estimation checks
- The same quantity can be written as a fraction, decimal, percent, or scientific notation — convert to one form before ordering or operating
- Positive integer exponents mean repeated multiplication (for example 3^4 = 3 × 3 × 3 × 3 = 81); apply exponents before multiplication and division in a mixed expression
- Common traps mix percent-of with percent-more-than, and confuse digit order with magnitude when decimals or scientific notation appear
Why quantity and operations matter on QL
Quantitative Literacy on the National Benchmark Test (AQL) measures whether you can handle numbers the way first-year study expects: compare sizes, convert representations, and compute accurately without a calculator. Formulae are supplied when a specialised relationship is needed, but ordinary arithmetic — including fractions, percentages, decimals, scientific notation, and positive exponents — is assumed.
This section builds the core quantity skill from the CEA QL construct: order quantities; estimate and compute with whole numbers, fractions, decimals, and percentages; use scientific notation; add, subtract, multiply, and divide; and raise numbers to positive integer powers. Later QL sections (ratios, multi-step chains, rates, and data) all rest on this fluency.
One quantity, many writings
Before you compare or calculate, rewrite every option in the same form. The conversion table below is worth memorising because QL stems deliberately mix forms.
| Fraction | Decimal | Percent | Scientific notation (where useful) |
|---|---|---|---|
| 1/2 | 0.5 | 50% | 5 × 10^(−1) |
| 1/4 | 0.25 | 25% | 2.5 × 10^(−1) |
| 1/5 | 0.2 | 20% | 2 × 10^(−1) |
| 1/8 | 0.125 | 12.5% | 1.25 × 10^(−1) |
| 3/8 | 0.375 | 37.5% | 3.75 × 10^(−1) |
| 2/5 | 0.4 | 40% | 4 × 10^(−1) |
| 3/5 | 0.6 | 60% | 6 × 10^(−1) |
| 3/4 | 0.75 | 75% | 7.5 × 10^(−1) |
| 7/8 | 0.875 | 87.5% | 8.75 × 10^(−1) |
Conversion rules (no calculator):
- Fraction → decimal: divide numerator by denominator (for example 3 ÷ 8 = 0.375).
- Decimal → percent: multiply by 100 (0.375 → 37.5%).
- Percent → decimal: divide by 100 (12.5% → 0.125).
- Percent → fraction: write over 100 and simplify (12.5% = 12.5/100 = 125/1000 = 1/8).
- Scientific notation: write as a × 10^n with 1 ≤ a < 10 (45 000 = 4.5 × 10^4; 0.0036 = 3.6 × 10^(−3)).
Worked example — ordering mixed forms
A residence office quotes four “occupancy buffers” for the same corridor: 0.45, 2/5, 47%, and 0.4. Which is largest?
Convert everything to decimals:
- 2/5 = 2 ÷ 5 = 0.4
- 47% = 47 ÷ 100 = 0.47
- 0.45 stays 0.45
- 0.4 stays 0.4
Ordered from smallest to largest: 0.4 = 2/5 < 0.45 < 0.47. So 47% is largest. The trap is treating “0.45 looks bigger than 47%” without converting — 47% is already 0.47.
Worked example — percentage of a fee
Annual residence fees are R12 480. A bursary covers 15% of the fee. How much does the bursary pay?
Break 15% into friendly pieces:
- 10% of 12 480 = 12 480 ÷ 10 = 1 248
- 5% is half of 10% = 1 248 ÷ 2 = 624
- 15% = 1 248 + 624 = R1 872
Check by another route: 15% = 3/20, so (12 480 ÷ 20) × 3 = 624 × 3 = 1 872. Same result.
Worked example — fraction of enrolment
A faculty enrols 2 400 first-years. Commerce takes 3/8 of them. How many commerce students is that?
- 2 400 ÷ 8 = 300
- 300 × 3 = 900
So commerce has 900 students. As a percent: 3/8 = 0.375 = 37.5% of the intake.
Worked example — scientific notation and addition
A campus water report lists storage as 2.4 × 10^3 litres in Tank A and 600 litres in Tank B. Combined volume?
- 2.4 × 10^3 = 2.4 × 1 000 = 2 400
- 2 400 + 600 = 3 000
- 3 000 = 3 × 10^3 litres
Do not add the coefficients 2.4 and 600 without aligning place value — that is a frequent QL trap.
Positive exponents and order of operations
A positive integer exponent means repeated multiplication of the base:
- 2^5 = 2 × 2 × 2 × 2 × 2 = 32
- 3^4 = 3 × 3 × 3 × 3 = 81
- 10^3 = 1 000
In a mixed expression, apply exponents before multiplication and division, and those before addition and subtraction (brackets first if present).
Example: Evaluate 12 + 8 × 3^2 ÷ 4.
- Exponent: 3^2 = 9
- Multiplication and division left to right: 8 × 9 = 72, then 72 ÷ 4 = 18
- Addition: 12 + 18 = 30
If you add 12 + 8 first, you get a wrong 45 — classic order-of-operations error.
Estimation as a safety net
Because calculators are banned, estimate before fine arithmetic. For load shedding: if a suburb loses power 4.5 hours in a 24-hour day, the share is 4.5 ÷ 24.
- 4.5 ÷ 24 = 45 ÷ 240 = 9 ÷ 48 = 3/16
- 3/16 = 0.1875 = 18.75% (about 19% of the day)
If an option says “about half the day,” reject it immediately — 18.75% is nowhere near 50%.
Common quantity traps
| Trap | Wrong move | Correct move |
|---|---|---|
| Ordering 0.09 vs 0.1 | “0.09 has more digits so it is larger” | 0.09 = 9/100 < 0.1 = 10/100 |
| 20% of vs 20% more than | Treat both as × 0.20 | “of” → × 0.20; “more than” → × 1.20 |
| Scientific notation | Add coefficients ignoring powers of 10 | Convert to ordinary form or match exponents first |
| Mixed operations | Left-to-right ignoring exponents | Brackets → exponents → ×÷ → +− |
| Fraction of a total | Multiply by denominator | Divide by denominator, then multiply by numerator |
Practice habit for this skill
- Convert every competing value to one form (usually decimals or percentages).
- Write each arithmetic step on paper — QL rewards process accuracy under time pressure.
- Estimate a sensible range, then compute exactly.
- Express the final answer in the form the stem requests (fraction, percent, or scientific notation).
Mastering these moves makes ratios and multi-step items in the next sections far less expensive in time.
Which quantity is the largest: 3/5, 0.58, 61%, or 7/12?
A registration fee is R3 200. After a 12.5% rebate, how much does the student pay?
A campus tank holds 1 600 litres of water. Students use 3/5 of the tank. How many litres remain?
Evaluate 2.4 × 10^3 + 6 × 10^2 and express the sum in scientific notation.