SAT — mathHard4 min read

Multi-Step Percent and Compound Interest

Solve compound interest and successive percentage change problems for the Digital SAT. Master the compound growth formula.

Tutor AI Team
Tutor AI Team

Compound interest and successive percentage changes involve applying percentage changes repeatedly over time. Unlike simple interest (where you earn interest only on the original amount), compound interest earns interest on previously earned interest. The Digital SAT tests this through financial contexts, population models, and multi-step percentage problems.

Core Concepts

Compound Interest Formula

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

  • PP = principal (starting amount)
  • rr = annual interest rate (as a decimal)
  • nn = number of times compounded per year
  • tt = number of years
  • AA = amount after tt years

Common Compounding Frequencies

Frequency nn
Annually 1
Semi-annually 2
Quarterly 4
Monthly 12
Daily 365

Successive Percentage Changes

Apply each percentage change sequentially using multipliers:

A 20% increase followed by a 10% decrease:

Final=Original×1.20×0.90=Original×1.08\text{Final} = \text{Original} \times 1.20 \times 0.90 = \text{Original} \times 1.08

Net effect: 8% increase (not 10%!).

The "Not 10%" Trap

A 20% increase then 20% decrease does NOT return to the original:

100×1.20×0.80=96100 \times 1.20 \times 0.80 = 96 — a net 4% decrease.

Simple vs. Compound Interest

  • Simple: A=P(1+rt)A = P(1 + rt) — interest only on original.
  • Compound: A=P(1+r/n)ntA = P(1 + r/n)^{nt} — interest on interest.

Strategy Tips

Tip 1: Use the Multiplier Method

For each percentage change, multiply by the appropriate factor. Chain them for successive changes.

Tip 2: Don't Add Percentages

A 10% increase followed by a 10% increase is NOT a 20% increase. It's 1.10×1.10=1.211.10 \times 1.10 = 1.21 = 21% increase.

Tip 3: Identify the Compounding Frequency

Read carefully: "compounded monthly" means n=12n = 12, not n=1n = 1.

Tip 4: Use the Calculator

For compound interest calculations, use the SAT calculator for large exponents.

Worked Example: Example 1

Problem

$10,000 at 6% compounded annually for 5 years.

A = 10000(1.06)^5 = 10000 \times 1.3382 = \13{,}382.26$

Solution

Worked Example: Example 2

Problem

$5,000 at 4% compounded quarterly for 3 years.

A = 5000(1 + 0.01)^{12} = 5000(1.01)^{12} = 5000 \times 1.1268 = \5{,}634.13$

Solution

Worked Example: SAT-Style

Problem

A shirt's price increases 15% then decreases 10%. What is the net percentage change?

1.15×0.90=1.0351.15 \times 0.90 = 1.035. Net: 3.5% increase.

Solution

Worked Example: Example 4

Problem

A population of 50,000 grows 3% per year. What is the population after 10 years?

P=50000(1.03)10=50000×1.3439=67,196P = 50000(1.03)^{10} = 50000 \times 1.3439 = 67{,}196

Solution

Worked Example: Example 5

Problem

An investment doubles in value. If it grew at 7% annually (compounded), approximately how many years did it take?

(1.07)t=2(1.07)^t = 2. Using the Rule of 72: 72/7≈10.372 / 7 \approx 10.3 years.

Solution

Practice Problems

  1. Problem 1

    $8,000 at 5% compounded annually for 4 years.

    Problem 2

    A price increases 25% then decreases 25%. What is the net change?

    Problem 3

    $2,000 at 3% compounded monthly for 2 years.

    Problem 4

    A stock goes up 10%, then up 20%, then down 15%. Net change?

    Problem 5

    How long for $1,000 to grow to $1,500 at 8% compounded annually?

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Common Mistakes

  • Adding successive percentages. 10% up then 10% down ≠ 0% change. It's a net decrease.
  • Confusing rr and r/nr/n. The rate PER PERIOD is r/nr/n, not rr.
  • Using simple interest formula for compound interest. They give different results.
  • Wrong exponent. The exponent is ntnt (periods × years), not just tt.

Key Takeaways

  • ✓

    Compound interest: A=P(1+r/n)ntA = P(1 + r/n)^{nt}.

  • ✓

    Successive changes: multiply the factors, don't add percentages.

  • ✓

    x%x\% increase then x%x\% decrease ≠ no change — it's always a net decrease.

  • ✓

    Rule of 72: doubling time ≈ 72/r72 / r (where rr is the percentage rate).

  • ✓

    Always check if interest is simple or compound and the compounding frequency.

Tags:
#sat#math#problem-solving#percentages#compound-interest

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