5.3 Worked Problem Walkthroughs

5.3 Worked Problem Walkthroughs

Speed on MK comes from rehearsing complete solutions until the steps are automatic. Work each example below by hand, then check against the explanation.

Percentages

Problem: What is 35% of 80? Convert the percent to a decimal: 35% = 0.35. Multiply: 0.35 × 80 = 28. The template is part = whole × rate.

Percent change: A price rises from $40 to $50. Change = (50 − 40)/40 = 10/40 = 0.25 = 25% increase. Always divide by the original value, not the new one.

Ratios and proportions

Problem: A recipe uses flour and sugar in a 3:2 ratio. If 12 cups of flour are used, how much sugar? Set up 3/2 = 12/x, cross-multiply → 3x = 24 → x = 8 cups. Cross-multiplication turns any proportion into a one-step linear equation.

Fractions

Problem: Compute 2/3 + 1/4. Find the least common denominator (12): 8/12 + 3/12 = 11/12. For division, multiply by the reciprocal: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8.

Exponents in context

Problem: Simplify (x⁵)/(x²) · x. Subtract exponents for the division (x³), then add for the multiplication (x³ · x¹ = x⁴) → x⁴.

Geometry

Problem: A right triangle has legs of 6 and 8. Find the hypotenuse. Apply a² + b² = c²: 36 + 64 = 100, c = √100 = 10 (a scaled 3-4-5 triple).

Problem: A circle has radius 5. Find its area. πr² = π(25) = 25π (≈ 78.5).

Translation cheat sheet

Estimation as a filter

Before exact arithmetic, estimate to eliminate choices. If 35% of 80 must be 'a bit more than a third of 80,' any answer above 40 or below 20 is impossible — you may answer correctly without finishing the computation when the survivors are far apart. Reserve precise calculation for close choices.

Averages and weighted averages

The mean is the sum divided by the count. If three test scores are 80, 90, and 100, the mean is 270/3 = 90. Watch for the reverse question: 'A student needs a 90 average over four tests and has 85, 88, and 92 — what is needed on the fourth?' Required sum = 90 × 4 = 360; current sum = 265; needed = 95. The trick is realizing the average tells you the total, then subtracting what you already have.

Unit conversions and rates

Keep units consistent before computing. 90 minutes is 1.5 hours; 2 feet is 24 inches. For rate problems, rate = quantity ÷ time. A car covering 150 miles in 3 hours averages 50 mph. MK rarely buries rate in a story (that is Arithmetic Reasoning), but a clean rate computation can appear.

Roots and fractional exponents in context

Problem: Simplify √72. Factor out the largest perfect square: 72 = 36 × 2, so √72 = 6√2. Problem: Evaluate 8^(2/3). Read the denominator as a root and the numerator as a power: take the cube root of 8 (= 2), then square it → 4. Fractional exponents combine the root and power rules, and reading them in the right order — root first, then power — keeps the arithmetic small.

A worked multi-step example

Problem: If 2(x + 3) − 4 = 10, find x². Solve the linear equation first: distribute → 2x + 6 − 4 = 10 → 2x + 2 = 10 → 2x = 8 → x = 4. The question asks for x², so the answer is 16, not 4. This illustrates Trap 4 from the next section: the stem asked for a transformation of the solution, not the solution itself. Solve fully, then re-read what was requested before bubbling.

Mixture and combination thinking

Problem: A class has 12 boys and 18 girls. What fraction of the class is boys? Total = 30; boys = 12; fraction = 12/30 = 2/5. Reduce fractions to lowest terms before checking the answer list, because the test almost always lists the reduced form. Problem: If 3 pencils cost $0.45, what does one pencil cost? Divide: 0.45/3 = $0.15. Unit-rate questions reduce to a single division once you identify the per-unit quantity.

Geometry composites

Problem: A square has the same area as a circle of radius 4. Find the square's side length. Circle area = π(16) = 16π ≈ 50.27; side = √50.27 ≈ 7.1. When a problem links two shapes, set their shared quantity (area, perimeter) equal and solve. Problem: Find the area of an L-shaped region by splitting it into two rectangles and adding the parts — decomposition turns an unfamiliar figure into familiar ones.

Practicing the read-then-solve habit

The single biggest accuracy gain on MK comes from a one-second pause to ask 'what exactly is being requested?' before computing. Is it x, x², the sum of roots, the perimeter, or the area? Many MK items deliberately solve cleanly to a value that is not the final answer, then require one more step. Build the habit now: underline the demand, solve, then return to the underline before you choose. In timed practice, tag any miss caused by answering the wrong quantity so you can see how often this single habit would have saved points.

Self-test prompts

After each worked set, quiz yourself with these prompts: Can I name the procedure this item tested? Can I solve a structurally identical item with new numbers? Can I explain why each of the three wrong choices is wrong? If you cannot do all three, the topic is not yet exam-ready and needs another pass of focused, untimed practice before you fold it back into timed sets.