EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.3 · Percentages
On the paper: Percent questions appear on every real EST I paper — usually 4–6 of them
Big idea. A percent is a ratio out of 100. Turn every sentence into a multiplier — '×1.p' for an increase of p%, '×(1 − p/100)' for a decrease — and every percent question becomes one multiplication or one division.
1Reverse percentage: the RESULT is given, not the start ↩️
🍼 Baby steps
- 1
'increased by 22.5% TO $14,325' → x × 1.225 = 14,325
October 2020. The paper never says "the original was" — it hides it after "to".
- 2
x = 14,325 ÷ 1.225 = 11,693.88 ≈ $11,694
Divide by the multiplier — never subtract 22.5% from the result.
- 3
'decreased by 11% TO $49,840' → x × 0.89 = 49,840 → x = 56,000
June 2021. A decrease of p% is a multiplier of (1 − p/100).
- 4
'after a 45% discount, price is $5' → x × 0.55 = 5 → x = $9.09
HS Sample 2026. Same move — the discount IS the decrease.
Trap · never subtract the percent from the result
2Successive change: multiply the factors, never add the percents ✖️
🍼 Baby steps
- 1
prices raised 25%, then a 15% discount → 1.25 × 0.85 = 1.0625
October 2020. Overall change: +6.25%, not +25% − 15% = +10%.
- 2
price gets a 10% tax, then a 15% cut → 1.10 × 0.85 = 0.935
HS Sample 2026. $625 → $625 × 0.935 = $584.38.
- 3
vaccine +15%, then +10% two weeks later → 1.15 × 1.10 = 1.265
December 2021. Overall +26.5%, not +25%.
- 4
'by what % to UNDO a 20% discount?' → need ×1.25 to invert ×0.8, since 1 ÷ 0.8 = 1.25
June 2021. Undoing a decrease needs a BIGGER percent increase than the discount was.
🎛 Play with it · chain two percent changes
Pick a start value and two changes (increase or decrease). Watch the bar shrink/grow at each step, and compare the real overall % to the wrong answer you'd get by just adding the two percentages.
100 × (1 + 25/100) × (1 − 20/100) = 100 · overall change +0%
Adding the two percentages instead gives 105 — wrong whenever both changes are non-zero, because the second change acts on the result of the first, not on the original 100.
3Growth and decay: the multiplier raised to a power 📈
🍼 Baby steps
- 1
population 9.12M in 2011, +1% every year → 9.12 × (1.01)¹¹ in 2022
May 2021. 2022 − 2011 = 11 years — count the years, not the digits.
- 2
9.12 × 1.01¹¹ = 10.1749… ≈ 10.174896 million
Each year multiplies by the SAME factor 1.01 — that is what "compounds" means.
- 3
deposit today at 8% annual interest, compounded MONTHLY, for 5 years, to reach $10,000
December 2021. Monthly compounding: the yearly rate is split into 12, and so is the time.
- 4
x × (1 + 0.08/12)^(12×5) = 10,000 → x = 10,000 ÷ (1.006667)⁶⁰ ≈ $6,712.10
"Compounded monthly" means the rate per period is (annual rate ÷ 12) and the number of periods is (years × 12).
4Real EST questions, solved twice 📄
From 2018 to 2019, the amount in Julia's bank account increased by 22.5% to $14,325. To the nearest dollar, what was the initial amount in her bank account?
- A.$11,694
- B.$14,010
14,325 × 0.9784 — an arithmetic slip, not a real move.
- C.$11,102
14,325 minus 22.5% of 14,325 — subtracts the percent from the wrong base.
- D.$12,775
14,325 × 0.89 — uses an 11% decrease, not a 22.5% increase.
Safe way · ~25 s
- x × 1.225 = 14,325.
- x = 14,325 ÷ 1.225 ≈ 11,693.88 ≈ $11,694.
⏱ 90-second way · ~10 s
- 14,325 ÷ 1.225 = 11,693.88 → $11,694.
Answer: A
A local supermarket offered a discount of 15% on its items after the government raised all the prices by 25%. Overall, by what percentage were the original prices raised in this supermarket in particular?
- A.8.625%
25% − 15% + 15% × ... — a mixed-up manual combination.
- B.7.250%
Averages the two percentages instead of multiplying the factors.
- C.6.250%
- D.5.625%
1.25 × 0.85 computed with a rounding slip.
Safe way · ~25 s
- 1.25 × 0.85 = 1.0625.
- Overall change: +6.25%.
⏱ 90-second way · ~10 s
- 1.25 × 0.85 = 1.0625 → +6.25%. Never 25 − 15 = 10%.
Answer: C
By what percentage should we raise the price of a good to return to its initial price, knowing that it was subject to a 20% discount two weeks ago?
- A.15%
20% × 0.75 — no real route to this number.
- B.20%
The same percentage as the discount — but undoing a decrease always needs MORE.
- C.25%
- D.30%
20% + 10% — guessing a bigger number without computing the inverse.
Safe way · ~25 s
- After the discount, price = 0.8 × original.
- To invert: 1 ÷ 0.8 = 1.25 → a 25% increase.
⏱ 90-second way · ~10 s
- 1 ÷ 0.8 = 1.25 → +25%. The inverse of ×0.8 is ×1.25, never ×1.2.
Answer: C
How much money would you need to deposit today at 8% annual interest compounded monthly to have $10,000 in the account after 5 years?
- A.$6,500.50
Uses simple interest, not compound.
- B.$6,680.13
Compounds annually instead of monthly.
- C.$6,712.10
- D.$5,989.30
Uses 10 years instead of 5.
Safe way · ~50 s
- Monthly rate = 8% ÷ 12 = 0.6667%. Periods = 5 × 12 = 60.
- x × (1.006667)⁶⁰ = 10,000 → x = 10,000 ÷ 1.48987 ≈ $6,712.10.
⏱ 90-second way · ~20 s
- Present value = 10,000 ÷ (1 + 0.08/12)⁶⁰ ≈ $6,712.10.
Answer: C
A company sells electric scooters for $625 each. Last month, they decided to apply a 10% tax fee on all products. This month, they reduced the total price of a scooter by 15%. Approximately, how much does a scooter cost now?
- A.$531.20
Applies only the 15% cut to the original $625, skipping the tax.
- B.$584.40
- C.$593.70
Adds the 10% and subtracts the 15% instead of multiplying the factors.
- D.$687.50
625 × 1.10 — the tax only, forgetting the discount that followed.
Safe way · ~25 s
- 625 × 1.10 = 687.50 (after tax).
- 687.50 × 0.85 = 584.375 ≈ $584.40.
⏱ 90-second way · ~10 s
- 625 × 1.10 × 0.85 = 584.375 ≈ $584.40.
Answer: B
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1"Increased/decreased BY p% TO result" → original = result ÷ (1 ± p/100). Never subtract the percent from the result.
- 2Two changes in a row: multiply the two factors. Adding the percentages is always wrong once both changes are non-zero.
- 3To undo a decrease of p%, you need MORE than a p% increase: the inverse of ×(1 − p/100) is ×1/(1 − p/100).
- 4"Grows/shrinks by p% every period, for n periods" → original × (1 ± p/100)ⁿ.
- 5"Compounded monthly/quarterly" → divide the annual rate by the number of periods per year, and multiply the years by that same number.
- 6"x% of y% of a number" → multiply the number by the two decimals in a row: N × (x/100) × (y/100).
- 7From a table or chart: percentage of a category = that category's value ÷ the TOTAL of every category, ×100.
Take with you
- ▸Every percent sentence becomes one multiplier: increase → ×(1 + p/100), decrease → ×(1 − p/100).
- ▸A reverse-percentage question gives you the result — divide by the multiplier to find the original.
- ▸Successive changes and growth/decay both chain the SAME multiplier idea: multiply factors, or raise one factor to a power — never add percentages.