Compound Interest and Repeated Percent Change

8.5: Compound Interest and Repeated Percent Change. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 63 parts Practice Flashcards

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Compound interest

Objective: Evaluate repeated interest growth without deposits.

Prerequisite skills: growth factors, percentages, and exponent rules. A balance earning rate r per compounding period is multiplied by 1+r each period. With no deposits or withdrawals, the compound-interest model is P(1+r)^t. Interest earned is final balance minus principal. Common error: using simple interest P(1+rt) for a compounded model. Restrictions and interpretation: Use the stated compounding frequency and distinguish nominal annual rate from the per-period rate.

Formulas

$A=P(1+r)^t$

Worked examples

Try itA $2300 deposit earns 10% interest compounded annually. With no other transactions, what is the balance after 2 years?Show answer

Each year multiplies by 1.10. After two years A=2300(1.10)²=2300×1.21=$2783. Therefore the requested answer is 2783.

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Compound interest

Objective: Evaluate repeated interest growth without deposits.

Formulas

$A=P(1+r)^t$

Worked examples

Try itA $2300 deposit earns 10% interest compounded annually. With no other transactions, what is the balance after 2 years?Show answer

Each year multiplies by 1.10. After two years A=2300(1.10)²=2300×1.21=$2783. Therefore the requested answer is 2783.

Repeated percentage changes

Objective: Multiply distinct percent-change factors in order.

Successive percent changes use the current amount as the next base. Multiply their factors; adding the percentages is generally wrong. Equal increase and decrease rates do not cancel because their bases differ. Common error: adding a percent increase and decrease as if they had the same base. Restrictions and interpretation: Use the stated compounding frequency and distinguish nominal annual rate from the per-period rate.

Formulas

$A_final=A₀(1+r)(1−s)$

Worked examples

Try itA price of $1080 rises 25%, then the new price is discounted 20%. What is the final price?Show answer

The factors are 1.25 and 0.80. Their product is 1.00, so the final price is 1080×1.00=$1080. These particular unequal rates cancel multiplicatively. Therefore the requested answer is 1080.

Several compounding periods per year

Objective: Use the per-period interest rate and number of periods.

For nominal annual rate r compounded m times per year, each period uses r/m and t years contains mt periods. The annual effective rate is (1+r/m)^m−1, which need not equal r. Common error: using the annual rate in every subannual period. Restrictions and interpretation: Use the stated compounding frequency and distinguish nominal annual rate from the per-period rate. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$A=P(1+r/m)^(mt)$

Worked examples

Try itA $2700 deposit earns a nominal annual rate of 20%, compounded semiannually. What is the balance after one year with no transactions?Show answer

Semiannual means m=2 periods per year. The rate per period is 0.20/2=0.10, and one year contains 2 periods. Thus A=2700(1.10)²=$3267. Therefore the requested answer is 3267.

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