Lesson 15 ~40 min Unit 1 · Review +100 XP

Unit 1 Review

You have covered place value, integers, fractions, decimals, percentages, and ratios. Time to pull it all together. Every question here draws on something you have learned.

Capstone challenge: A recipe for 6 people needs flour and sugar in the ratio 3:2. You have $rac{3}{4}$ kg of flour. You need to increase the recipe by 25% for a dinner party. What percentage of your flour will be used? How much sugar do you need? This uses ratios, fractions, and percentages all at once.

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Whole Numbers & Integers

Lessons 1–3

Place value system: ones, tens, hundreds, thousands, millions. Integers include negative numbers. Adding a negative = subtracting. Subtracting a negative = adding. Use a number line to visualise.

$a + (-b) = a - b$ and $a - (-b) = a + b$

On a number line: right = bigger, left = smaller.

$|{-a}| = a$. Absolute value is always positive.

BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Fractions

Lessons 4–9

Equivalent fractions: multiply/divide both parts by the same number. Add/subtract: find LCD first. Multiply: straight across (cross-cancel first). Divide: keep, change, flip. Mixed numbers: convert to improper first.

$rac{a}{b} + rac{c}{d} = rac{ad + bc}{bd}$ (LCD method preferred)

$rac{a}{b} imes rac{c}{d} = rac{ac}{bd}$. No LCD needed. Cross-cancel first.

$rac{a}{b} \div rac{c}{d} = rac{a}{b} imes rac{d}{c}$. Keep, change, flip.

Simplify by dividing by HCF. Always simplify final answers.

Decimals

Lessons 10–11

Place value extends to the right: tenths, hundredths, thousandths. +/−: line up decimal points. ×: ignore dots, multiply, count total d.p., place dot. ÷: shift both dots to make divisor whole.

Line up dots for +/−. Add trailing zeros.

×: total d.p. in question = d.p. in answer.

÷: shift both equally. Dot goes above.

Always estimate first to check reasonableness.

Percentages

Lessons 12–13

% means per hundred. % → decimal: ÷100. Decimal → %: ×100. % of amount: (% as decimal) × amount. Increase: ×(1 + p/100). Decrease: ×(1 − p/100). GST: ×1.10 incl., ÷1.10 excl.

50% = $rac{1}{2}$ = 0.5, 25% = $rac{1}{4}$ = 0.25, 75% = $rac{3}{4}$ = 0.75

+p% means ×(1 + p/100). −p% means ×(1 − p/100).

Profit % = $rac{ ext{profit}}{ ext{cost}} imes 100$

GST reverse: never subtract 10%, divide by 1.10.

Ratios & Rates

Lessons 14

Ratio compares quantities with no units. Simplify by HCF. Same units first. Divide amount: total parts = sum, one part = amount ÷ parts. Rate: different units. Unit rate = total ÷ number of units.

Simplify: divide both sides by HCF.

Divide: one part = total ÷ sum of ratio parts.

Equivalent ratios: $rac{a}{b} = rac{c}{d}$ means $ad = bc$.

Unit rate = total ÷ quantity. Lower = better value.

Common Mistakes Across the Unit

watch out

Integer error: $5 - (-3) = 2$. No! Two negatives = positive. $5 - (-3) = 5 + 3 = 8$.

Fraction error: $rac{1}{2} + rac{1}{3} = rac{2}{5}$. No! Different denominators need LCD. $rac{3}{6} + rac{2}{6} = rac{5}{6}$.

Decimal error: $0.3 imes 0.2 = 0.6$. No! $3 imes 2 = 6$, 2 d.p. total, so $0.06$.

Percentage error: 20% of 80 = $20 imes 80 = 1600$. No! 20% = 0.20. $0.20 imes 80 = 16$.

Ratio error: 2:3 with $50 means $20 and $30 by luck. Use the method: parts = 5, each = $10.

GST error: $110 incl. GST = $110 - $10 = $100. No! $110 \div 1.10 = $100. Not subtract 10%.

How are you completing this lesson?

Mixed Quick Check · 5 questions (all topics)

12 ÷ 0.4 = ?

+10 XPDecimals

$\frac{3}{4} + \frac{1}{6} = ?$

+10 XPFractions

$220 incl. GST. Price before GST?

+10 XPPercentages

$180 in ratio 4:5. Larger share?

+10 XPRatios

$(-3) \times (-4) + (-2) \times 5 = ?$

+10 XPIntegers

Mixed Show Your Working · 3 questions (multi-topic)

Show Your Working

12 marks total

ApplyMedium4 MARKS

Q6. A shop buys an item for $80 and marks it up by 35% for the selling price. (a) What is the selling price? (b) In a sale, the shop offers 20% off the selling price. What is the sale price? (c) What is the overall percentage profit from the original cost to the sale price?

Answer in your workbook.

ApplyMedium4 MARKS

Q7. (a) Calculate $\frac{5}{6} - \frac{3}{8} + \frac{1}{4}$. Show the LCD and all conversion steps. (b) A recipe uses flour and sugar in the ratio 5:2. If you use $\frac{3}{4}$ kg of flour, how much sugar is needed? (c) Express your answer to (b) as a decimal.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A student calculated $2.4 \div 0.3 \times 0.5$ and got 4. Another student got 0.04. (a) Which student is correct? (b) Explain what mistake the other student made. (c) Write a general rule for the order of operations when decimals and division are involved.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — 12 ÷ 0.4 = 120 ÷ 4 = 30.

2. C — LCD(4,6) = 12. 9/12 + 2/12 = 11/12.

3. A — $220 ÷ 1.10 = $200.

4. D — Parts = 9, each = $20, 5×$20 = $100.

5. B — (-3)×(-4) = 12, (-2)×5 = -10, 12 + (-10) = 2.

Show Your Working Model Answers

Q6 (4 marks): (a) $80 × 1.35 = $108 [1]. (b) $108 × 0.80 = $86.40 [1]. (c) Profit = $86.40 − $80 = $6.40 [0.5]. Profit % = 6.40/80 × 100 = 8% [1.5].

Q7 (4 marks): (a) LCD(6,8,4) = 24 [0.5]. 20/24 − 9/24 + 6/24 = 17/24 [1.5]. (b) 5:2 means 5 parts flour = 2 parts sugar [0.5]. Sugar = 3/4 × 2/5 = 6/20 = 3/10 kg [0.5]. (c) 3/10 = 0.3 kg [1].

Q8 (4 marks): (a) Student 1 (answer = 4) is correct [1]. Working: 2.4÷0.3 = 8, 8×0.5 = 4 [0.5]. (b) Student 2 did 0.3×0.5 first = 0.15, then 2.4÷0.15 = 16, or misplaced the decimal [1]. (c) Division and multiplication have equal precedence — work left to right [1.5].

Capstone Challenge · +30 XP, +15 coins

The Ultimate Mix

A rectangular garden has length and width in the ratio 5:3. The perimeter is 64 m. (a) Find the actual length and width. (b) The owner wants to plant vegetables on exactly 40% of the garden area. What area is for vegetables? (c) Fertiliser costs $12.50 per 5 kg bag and covers 8 m² per kg. How many bags are needed for the whole garden, and what is the total cost? (d) GST is added to the fertiliser cost. What is the final amount?

Reveal solution

(a) Perimeter = 2(L+W) = 64, so L+W = 32. Ratio 5:3 means 8 parts = 32, so 1 part = 4. Length = 5×4 = 20 m, Width = 3×4 = 12 m. (b) Area = 20×12 = 240 m². 40% = 0.40×240 = 96 m². (c) Total area = 240 m². Coverage = 8 m²/kg. Need 240÷8 = 30 kg. Bags: 30÷5 = 6 bags. Cost: 6×$12.50 = $75. (d) $75 × 1.10 = $82.50.

Unit 1 Complete Review

Place Value

Ones, tens, hundreds, thousands

Integers

+, −, ×, ÷ with negatives

BODMAS

Order of operations

Fractions

+/− (LCD), × (across), ÷ (KCF)

Equivalent

× or ÷ both by same

Simplify

HCF of top and bottom

Decimals

Tenths, hundredths, thousandths

+ / −

Line up decimal points

× / ÷

Count d.p. / Shift dots

Percentages

% means /100

Multiplier

×1.p for +%, ×0.q for −%

GST

×1.10 incl, ÷1.10 excl

Ratios

Compare, no units

Simplify

÷ HCF both sides

Divide

Total ÷ parts = each

Rate

Per 1 unit

Unit 1 Key Formulas

revision sheet

$a - (-b) = a + b$ and $a + (-b) = a - b$

BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction

$\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$. LCD method preferred.

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$. Cross-cancel first.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. Keep, change, flip.

Decimals: line up dots (+/−), count d.p. (×), shift dots (÷)

%+: ×(1 + p/100). %−: ×(1 − p/100). GST: ×1.10 / ÷1.10

Profit % = $\frac{\text{profit}}{\text{cost}} \times 100$

Ratio: divide amount by total parts, multiply each ratio number

Unit rate = total ÷ quantity. Compare for best value.

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