Percent Applications

7.B.1Percent change01/03

Percent change — increase and decrease.

Step01of 03

Percent change formula.

% change = \frac{new - old}{old} \cdot 100%

Positive = increase, negative = decrease. Always divide by the ORIGINAL.

7.B.1 — Percent change — increase and decrease.

Step 01 of 03

Percent change formula.

% change = \frac{new - old}{old} \cdot 100%

Positive = increase, negative = decrease. Always divide by the ORIGINAL.

Step 02 of 03

Worked example. A shirt's price went from $40 to $50.

\frac{50 - 40}{40} = \frac{10}{40} = 0.25 = 25% increase

Step 03 of 03

If price drops $50 → $40, percent change is $(40 - 50) /50 = - 20%$ — a 20% decrease (NOT 25%, even though the dollar change is the same). The denominator changes.

Key insight

Always divide by the ORIGINAL value. Up and down by the same dollar amount give different percentages because the original is different.

7.B.2 — Discount, markup, tax.

Step 01 of 03

Discount = price reduction. Compute discount amount, then subtract.

discount = original \cdot rate

sale price = original - discount

$80 jacket, 25% off: discount $= 80 \cdot 0.25 = 20$ ; sale $= 80 - 20 = 60$ .

Step 02 of 03

Markup / tax = price addition. Same idea, just add.

tax = price \cdot rate, total = price + tax

$60 with 8% tax: tax $= 60 (0.08) = 4.80$ ; total $= 64.80$ .

Step 03 of 03

One-step shortcut. Combine the multiplication.

25% off	→multiply by $(1 - 0.25) = 0.75$
8% tax	→multiply by $(1 + 0.08) = 1.08$
Combined: 25% off then 8% tax	→ $price \cdot 0.75 \cdot 1.08$

Key insight

Discount = subtract, markup/tax = add. The "1 ± rate" multiplier collapses two steps into one — useful when chaining multiple percent changes.

7.B.3 — Tip and commission.

Step 01 of 03

Both work like a tax — multiply the relevant amount by a rate.

tip = bill \cdot tip rate

commission = sale \cdot rate

Step 02 of 03

Restaurant bill $48, 18% tip: $48 \cdot 0.18 = $8.64$ . Total $= 48 + 8.64 = $56.64$ .

Salesperson sells $12,000 of furniture, earns 6% commission: $12000 \cdot 0.06 = $720$ .

Step 03 of 03

"Tip on the pre-tax amount" vs. "tip on the total" depend on local custom — read the question. Same arithmetic either way; just different base.

Key insight

Tip and commission are percent-of-base calculations. Pay attention to WHICH base the question uses (pre-tax bill, total bill, gross sale, net sale, etc.).

7.B.4 — Simple interest.

Step 01 of 03

Simple interest formula.

I = P \cdot r \cdot t

$P$ = principal, $r$ = annual rate (decimal), $t$ = time in years. Total amount = $P + I$ .

Step 02 of 03

Worked example. $2,000 at 4% for 3 years.

I = 2000 \cdot 0.04 \cdot 3 = $240

A = 2000 + 240 = $2240

Step 03 of 03

Watch units. If rate is annual but time is in months, convert: 6 months = 0.5 years. If rate is monthly, multiply by months directly.

Key insight

I = P r t

. Simple interest grows LINEARLY (proportional to time), unlike compound interest which grows exponentially. Always check that

r

and

t

use matching time units.

7.B.5 — Percent error.

Step 01 of 03

Percent error = how far an estimate or measurement is from the true value, as a percent of the truth.

% error = \frac{∣ estimate - actual ∣}{actual} \cdot 100%

Always positive (absolute value), and always divides by ACTUAL.

Step 02 of 03

Predicted attendance: 250. Actual: 280.

% error = \frac{∣250 - 280∣}{280} = \frac{30}{280} \approx 10.7%

Step 03 of 03

Don't confuse percent error with percent change. Percent change uses the ORIGINAL as the denominator (and keeps sign). Percent error uses the ACTUAL/TRUE value (and is always positive).

Key insight

Percent error always divides by the actual value and uses absolute value. It measures accuracy — how far off was your guess?

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7.B.1 – 7.B.5 · CCSS 7.RP.A.3 · Full curriculum →