Percent–Change, Price–Quantity & Income–Expenditure Cheat-Sheet

Core Formulas, Rules & Short‐cuts

• Relation between money variables
– Expenditure / Revenue / Cost: E = P \times Q
– When one factor changes by (+a\%) and the other by (+b\%) (use (−b\%) for a fall), combined % change in the product:
\text{Net\%} = a + b + \frac{ab}{100}

• Successive change on the same quantity (e.g., price up then down):
– Convert each % to a multiplier ((+20\% \Rightarrow 1.20), (−15\% \Rightarrow 0.85)), multiply all multipliers, then reconvert to %.
– Shortcut for exactly equal rise & fall of (x\%): net fall = \frac{x^2}{100}\%

• Consumption vs. price change (keeping expenditure fixed):
\frac{\text{New quantity}}{\text{Old quantity}} = \frac{100}{100 + a} if price rises by (a\%).
– Example: price (+25\%) ⇒ quantity must fall to \frac{100}{125}=0.8 (i.e., −20 %).

• Income – Expenditure – Savings diagram
– Always normalise: let income = 100.
– Savings = 100 − expenditure %. Any % change questions can now be handled with the basic product formula.

• Converting mixed fractions or awkward percentages
– 6\tfrac14\% = 6.25\% = \frac1{16}
– 14\tfrac{2}{7}\% = \frac{1}{7} (handy for price–quantity problems because of clean reciprocals).

Single–Concept Illustrations

1. Rice with Unexpected Guests

• Family of 4 usually buys a fixed quantity (say 1 unit).
• Price rises by 12.5\% \;(=1\/8) \Rightarrow \text{price multiplier}=1.125
• Two guests arrive and together eat \tfrac25 of the family’s normal consumption ⇒ total quantity multiplier =1+\tfrac25 = 1.4
• Net expenditure change
1.125 \times 1.4 = 1.575 \Rightarrow 57.5\% \text{ rise}
(Many classroom solutions mistakenly report 2.5 %; correct is 57.5 %).

2. Sugar – Price ↑ 32 %, Expenditure ↑ 10 %

• Price multiplier = 1.32
• Expenditure multiplier = 1.10
• Required quantity multiplier = \frac{1.10}{1.32}=0.8333 ⇒ new quantity \approx 8.33\,\text{kg} if original was 10 kg.

3. Sugar – Price ↑ 28 %, keep expenditure +12 %

• Multiplier method: \frac{1.12}{1.28}=0.875 ⇒ drop quantity by 12.5\% (≈ 8.75 kg when starting from 10 kg).

4. Bus Fare – Fare ↓15 %, Ridership ↑40 %

• Use combined rule with opposite signs:
−15 + 40 + \frac{(−15)(40)}{100}=25−6=19\%
⇒ Revenue rises 19 %.

5. Books – Price ↓10 %, Volume ↑35 %

• Net revenue: −10+35+\tfrac{(−10)(35)}{100}=25−3.5=21.5\% increase.

6. TVs – Price ↓10 %, Volume ↑30 %

• Net revenue: −10+30−3=17\% rise.

7. Price ↓15 %, Volume ↑35 %

• Net receipt =−15+35−5.25=14.75\% rise.

8. Successive Rises: +20 % then +15 %

• Multiplier = 1.20 × 1.15 = 1.38 ⇒ 38 % overall rise (not 35 %).

9. +10 % then –5 %

• Multiplier = 1.10 × 0.95 = 1.045 ⇒ 4.5 % net rise.

10. Double Discount on Books (–20 % list ⇒ regular, then –15 % sale)

• Multipliers: 0.80 then 0.85 ⇒ 0.68 ⇒ total markdown 32 % off list.

11. Rectangle – L ↑40 %, B ↓20 %

• Net area: +40−20−8=+12\% larger.

12. Import Duty & Supplementary Tax

• Duty = 50 % of cost ((C)), sup-tax = 30 % of (C + duty)
⇒ Duty = 0.5C , Base for tax = 1.5C , tax = 0.3×1.5C = 0.45C.
• Total = 0.95C = 28 500 ⇒ C=30\,000.

13. Bank Deposit with Levy & Interest Tax

• Net amount received: 55 280.
– Principal 50 000 ⇒ net interest actually pocketed = 5280.
– Before 10 % tax on interest: gross interest =\frac{5280}{0.9}=5866.67
– Total levy = 1.20 (lump sum).
– Gross simple-interest rate: \frac{5866.67}{50\,000}\times100≈11.73\%\approx12\%.

14. Raw Material & Labour Mix

• Raw price ↑15 %; labour share rises from 25 % to 30 % of raw cost.
• Let old raw = 100, old labour = 25, total = 125.
• New raw (after wished-for cut) = x, labour = 0.30x. Need new total = 125.
Equation: x + 0.30x = 1.15×100 + 0.30×100 = 145
⇒ 1.30x = 145 ⇒ x = 111.54
• Raw must fall from 115 (the would-be cost with no cut) to 111.54 ⇒ \frac{115−111.54}{115}=3\% ≈ 3.0 % reduction in usage.

Income–Expenditure–Savings Problems

Generic Table (initial income = 100)

Key Classroom Questions

1. Spend 75 % of income. Income ↑20 %, expenses ↑10 %.
   – Old savings = 25.
   – New income = 120, new expenses = 82.5 ⇒ savings = 37.5.
   – % rise in savings: \frac{37.5−25}{25}\times100=50\%.

2. Same set-up but expenses ↑15 % ⇒ new savings
   – Expenses = 86.25, savings = 33.75 ⇒ rise =35\%.

3. Fixed decision: always save 10 % of income. Income ↑25 % ⇒ savings ↑25 % (trivial proportionality).

4. Robin: spends 80 % (so saves 20 %). Income ↑12 %, savings ↓10 % (from 20 down to 18).
   – Let old income = 100 (spends 80). New income 112, new savings 18 ⇒ expenditure 94.
   – % change in expenditure: \frac{94−80}{80}=17.5\% rise.

5. Russell: saved 20 % in 2016. In 2017 no raise, but expenditure ↑20 %.
   – Old income = S, old save = 0.2S, old spend = 0.8S.
   – New spend = 0.8S×1.2 = 0.96S, saving = S − 0.96S = 0.04S.
   – Given new saving 1000 ⇒ 0.04S = 1000 ⇒ salary =25\,000\,/\text{month}.

6. Ritu saves x %. Income ↑26 %, expenditure ↑20 %, savings ↑50 %.
   – Work with fraction savings = x/100.
   – Equation: \frac{(100−x)\times1.20}{100\times1.26}=\frac{1−x/100}{1.26}=\frac{(100−x)1.20}{100(1.26)} ⇒ solving gives x=20.

7. Comparative income: A spends 60 %, B spends 75 %. Savings of A are 20 % higher than B’s.
   – Let B’s income = 100, spends 75, saves 25.
   – A’s saving = 30 ⇒ income 75, saving 30 ⇒ spend 45 ⇒ savings indeed 20 % more.
   – Income of A less than B: \frac{100−75}{100}=25\%.

Price-Drop & Extra-Quantity Type Problems

Miscellaneous Quick Facts

• If price is cut exactly 40 %, you can buy \tfrac{1}{0.6}=\tfrac53 times the old quantity.
• "Buy 8 for the price of 5" sale ⇒ effective discount =\frac{8−5}{8}=37.5\%.
• Magazine problem: revenue constant ⇒ P0 Q0 = 1.20 P0 (Q0−4) ⇒ Q_0=24 subscribers last year.