Number and Quantity
The Number and Quantity domain tests a teacher’s command of number systems, operations, proportional reasoning and the precision needed to explain them. Questions often ask you to choose an efficient representation, justify a procedure, or diagnose an error in place value, fractions or percentages.
Core rules and concepts
- The base-ten system is positional: the value of a digit depends on its place, and regrouping trades one unit of a higher place for ten of the next lower place.
- Integers are closed under addition, subtraction and multiplication but not division; division by zero is undefined.
- A fraction a/b is meaningful when b ≠ 0. Equivalent fractions are formed by multiplying numerator and denominator by the same nonzero number.
- Percent means ‘per hundred’: p% = p/100. Percentage change is (new − old)/old × 100%, using the original amount as the base.
- Ratio compares quantities of the same kind; a proportion states two ratios are equal. Direct variation has constant quotient y/x; inverse variation has constant product xy.
- HCF is the product of common prime factors; LCM is the product of the highest powers of all primes present.
Worked example
A class has 24 girls and 16 boys. Express the ratio of girls to boys in simplest form, then find the percentage of girls. Ratio = 24:16. Divide both terms by HCF 8: 3:2. Total = 40, so girls’ share = 24/40 = 0.6 = 60%. The ratio 3:2 does not mean girls are 3/2 of the class; it compares girls with boys.
Traps that cost marks
- Using the new amount as the base in percentage change instead of the original.
- Cancelling across addition, as in (x+2)/(x+4), instead of cancelling common factors.
- Writing 0.25% as 0.25 rather than 0.0025.
- Confusing HCF with LCM when simplifying ratio or adding fractions.
- Treating a ratio part as a fraction of the whole without checking the number of parts.