Fractions, decimals & percentages

Fluency

Fluency

$5$ (because $0.762 \geq 0.5$ ).
$12.4$ (because the hundredths digit is $8$ ).
$0.05$ (because the thousandths digit is $5$ ).
$18.74.
$-\dfrac{2}{3}$ is smaller. It sits further to the left of zero on the number line.
$-1.5,\ -\dfrac{3}{4},\ 0,\ 0.25,\ \dfrac{3}{2}$ - already in order.
Many answers. Examples: $-0.25$ , $-\dfrac{1}{4}$ , $-\dfrac{1}{3}$ .

Reasoning

$\dfrac{2}{3},\ 0.68\,(68\%),\ 0.7,\ \dfrac{7}{10}$ . Note $\dfrac{7}{10} = 0.70 = 0.7$ , so these are equal. Correct order: $\dfrac{2}{3},\ 68\%,\ 0.7 = \dfrac{7}{10}$ .
$\dfrac{11}{12}$ . Method: common denominator $12$ ; $\dfrac{8}{12} + \dfrac{9}{12} - \dfrac{6}{12} = \dfrac{11}{12}$ .
$\dfrac{1}{2}$ . Method: $\dfrac{1}{4} + \dfrac{1}{4} = \dfrac{2}{4}$ .
$\dfrac{1}{2}$ cup. Method: $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{1}{2}$ .
$9$ . Method: $\dfrac{9}{12} = \dfrac{3}{4}$ .
$45\%$ . Method: $\dfrac{18}{40} \times 100$ .
$45\%$ . Method: $\dfrac{27}{60} \times 100$ .
$68. Method: $85 \times 0.80 = 68$ .
$336. Method: $320 \times 1.05 = 336$ .
$60$ . Method: $x \times 1.40 = 84$ , so $x = 84 \div 1.40$ .
$36$ . Method: $\dfrac{2}{5} \times 120 = 48$ , then $\dfrac{3}{4} \times 48 = 36$ .
$\dfrac{3}{8}$ . Method: $\dfrac{8}{8} - \dfrac{3}{8} - \dfrac{2}{8}$ .

Reasoning

Sam is wrong. The correct answer is $\dfrac{5}{6}$ . Fractions must share a denominator before you can add them: $\dfrac{1}{2} = \dfrac{3}{6}$ and $\dfrac{1}{3} = \dfrac{2}{6}$ , so $\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$ . You cannot add the tops and bottoms separately.
Dividing by a fraction means multiplying by its reciprocal. The reciprocal of $\dfrac{1}{2}$ is $2$ , so $n \div \dfrac{1}{2} = n \times 2$ . Concretely, asking “how many halves fit in $n$ ?” gives twice as many as whole units, i.e. $2n$ .
Not the same. Starting from $100: $50\%$ off gives $50; then $20\%$ off $50 gives $40. A flat $70\%$ off $100 would leave $30. Because percentages compound on the new running total, the combined discount here is only $60\%$ .
Yes, $\dfrac{7}{13}$ is greater than $\dfrac{1}{2}$ . Half of $13$ is $6.5$ , and $7 > 6.5$ , so seven thirteenths is more than half.
$0.3 = \dfrac{3}{10}$ , not $\dfrac{1}{3}$ . The fraction $\dfrac{1}{3} = 0.333\ldots$ (the $3$ s repeat forever), so $0.3$ is slightly less than $\dfrac{1}{3}$ .

Problem solving

$70.20. Method: $65 \times 1.08$ .
$\dfrac{3}{8}$ . Method: $1 - \dfrac{5}{8}$ .
$60\%$ walk; $40\%$ do not. Method: $\dfrac{18}{30} \times 100 = 60$ .
$25\%$ off. Method: discount $60; $\dfrac{60}{240} \times 100$ .
$40. Method: $\dfrac{1}{5} \times 25 = 5$ per week; $\times 8$ .
$480$ litres. Method: $\dfrac{3}{4} - \dfrac{1}{2} = \dfrac{1}{4}$ of the tank is $120$ L, so full tank is $4 \times 120$ .
$35. Method: $x \times 1.20 = 42$ , so $x = 42 \div 1.20$ .
$840$ students. Method: $750 \times 1.12$ .

Tier 1: basic skills