3.2 Core Problem Types and Solution Workflows

3.2 Core Problem Types and Solution Workflows

Most AR items belong to a handful of families. Recognizing the family in the first 15 seconds tells you which formula to write down, which collapses a word problem into a fill-in-the-blank.

Percent and percent change

The master relationship is part = percent × whole. For change problems use percent change = (new − old) ÷ old × 100.

• "Of" means multiply: 18% of 250 = 0.18 × 250 = 45.
• Percent increase: new = old × (1 + rate). A $40 item up 15% = 40 × 1.15 = $46.
• Percent decrease: new = old × (1 − rate). $40 down 15% = 40 × 0.85 = $34.
• Reverse percent trap: a price raised 20% then cut 20% does not return to the start. $100 → $120 → $96. The two percents are taken from different bases.

Distance, rate, and time

Every DRT problem is d = r × t. Identify which of the three the stem leaves blank, then solve.

For average speed over a round trip, do not average the two speeds — use total distance ÷ total time. Driving 60 mi out at 60 mph (1 hr) and back at 30 mph (2 hr) gives 120 mi ÷ 3 hr = 40 mph, not 45.

Work-rate problems

When two workers or machines combine, add their rates, not their times. If A finishes a job in a hours and B in b hours, the combined rate is 1/a + 1/b, and total time is the reciprocal of that sum. Pump A fills a tank in 4 hr (rate 1/4) and pump B in 6 hr (rate 1/6); combined rate = 1/4 + 1/6 = 3/12 + 2/12 = 5/12 tank per hour, so total time = 12/5 = 2.4 hours. The trap answer 5 hours comes from averaging the two times.

Interest: simple vs. compound

Simple interest is I = P × r × t — interest only on the original principal. $1,000 at 5% for 3 years = 1000 × 0.05 × 3 = $150 interest. Compound interest earns interest on interest: A = P(1 + r)ᵗ. The same $1,000 at 5% compounded annually for 3 years = 1000 × 1.05³ ≈ $1,157.63, so about $157.63 interest. On a 29-minute, no-calculator subtest, AR favors simple interest or short compounding (2–3 periods you can multiply by hand).

Ratios, proportions, and unit rates

Set up a proportion and cross-multiply: if 3 filters cost $12, then 7 filters cost x, so 3/12 = 7/x → 3x = 84 → x = $28. A unit rate reduces a quantity to "per one" — 360 miles on 12 gallons = 30 miles per gallon — which makes comparison and scaling trivial.

Averages and weighted averages

Simple average = sum ÷ count. A weighted average multiplies each value by its weight (count or frequency) before summing. If 4 airmen score 90 and 6 score 80, the average is not 85; it is (4×90 + 6×80) ÷ 10 = (360 + 480) ÷ 10 = 84. Weighting by the larger group pulls the mean toward 80.

The workflow is identical across families: (1) name the family, (2) write its formula, (3) substitute with unit-checked numbers, (4) solve, (5) confirm the answer matches the quantity asked. Skipping step 5 is the most common reason a confident calculation lands on a distractor.

Two more families worth pre-loading

Mixture problems combine quantities at different concentrations or prices, and they reduce to a weighted total. To mix a 10% saline solution with a 30% solution to get 20 liters of 25%, set up total solute: 0.10x + 0.30(20 − x) = 0.25 × 20. Solve: 0.10x + 6 − 0.30x = 5 → −0.20x = −1 → x = 5 liters of the 10% solution and 15 liters of the 30%. The same setup handles blended coffee prices or alloy compositions.

Basic probability appears as "what is the chance" stems: probability = favorable outcomes ÷ total outcomes. Drawing one defective part from a bin of 3 defective and 17 good is 3 ÷ 20 = 0.15, or 15%. For two independent events both occurring, multiply the probabilities; for either-or with no overlap, add them.

Choosing the formula fast

Keep these seven families and their first moves on a single index card. On test day you cannot bring it, but the act of rebuilding the card from memory each morning during prep cements the formulas so that recognition — the slowest step for most candidates — becomes automatic and the 70-second clock stops being the enemy.