MA1 Integrating Essential Skills

Lesson

ACT groups about a fifth of the scored Math questions under Integrating Essential Skills, and ACT's own framing of the category is the useful part: the content is material you met before high school — rates, proportions, percentages, averages, area and volume, expressing a number in more than one form — but the questions apply it in ways that take two or three steps and rarely look like the exercise you first learned it from. Almost nothing here is hard to understand. The difficulty is in not losing the thread halfway through, and in noticing which familiar operation an unfamiliar sentence is actually asking for.

That means these are rarely the questions you get wrong because you did not know something. They are the questions you get wrong because you answered a slightly different question than the one on the page — you found the discount instead of the price, the total instead of the difference, the area instead of the perimeter. Read the last sentence of the problem twice, and underline the units it asks for.

Rates: the units are the instructions

A rate is a fraction with units on the top and the bottom, and a unit conversion is just multiplying by a fraction equal to 1. Set the chain up so the unit you want to be rid of appears once on top and once on the bottom, and it cancels:

• 45 miles/hour × 1 hour/60 minutes = 0.75 miles/minute. Hours cancelled because one was on top and one was on the bottom.
• If you had multiplied by 60 minutes/1 hour instead, the answer would carry units of miles·minutes per hour squared — nonsense, and visibly nonsense. That is the check.
• Rates arrive in odd units on purpose: dollars per square foot, gallons per 100 miles, pages per minute, liters per hectare. The odd unit is not a harder idea, only an unfamiliar one. Write it as a fraction and it behaves like every other rate.
• Whether to multiply or divide is never a guess. Ask which arrangement cancels the unit you want to lose.

Percent change is measured against the original

Every percent-change question is the same fraction:

percent change = (new − original) ÷ original × 100

The denominator is the original — the amount you started with, the "before", the old price — every single time. A price that goes from $40 to $50 rose by 10/40 = 25%. Coming back down from $50 to $40 is a fall of 10/50 = 20%. Same $10, different percentages, because the starting points differ. Dividing by the new amount is the single most common error in this category.

The same fraction run backwards handles reverse percent questions, the ones that give you the price after the change and ask what it was before. If a price rose 20% to reach $54, then $54 = 1.20 × original, so the original is 54 ÷ 1.20 = $45. Taking 20% off $54 gives $43.20, which is not the answer to any question that was asked.

Two 10% changes are not a 20% change

Percent changes multiply; they do not add. A $200 coat marked up 10% and then marked down 10% is not back where it started:

• 200 × 1.10 = 220, then 220 × 0.90 = 198. A 1% net loss, not a wash.
• Two successive 10% increases give 1.10 × 1.10 = 1.21 — a 21% increase, not 20%. The extra 1% is the second increase acting on the first increase.
• Compound interest is this same idea with a period attached: A = P(1 + r)^t, against simple interest's A = P(1 + rt).
• A discount and a sales tax do commute — 0.80 × 1.07 and 1.07 × 0.80 are the same number — but neither equals subtracting 7 from 20 and taking 13% off.

Average speed is total distance over total time

Average speed is not the average of the speeds. It is one division: every mile travelled, divided by every hour spent. The two agree only when the time spent at each speed is equal, which is almost never the case in a problem that bothers to give you two speeds.

Weighted averages behave the same way. Two classes averaging 78 and 90 give a combined average of 84 only if the classes are the same size. With 30 students at 78 and 10 at 90, the combined average is (30×78 + 10×90) ÷ 40 = 81. Multiply each average by its count, add, then divide by the total count.

Scale a length, and area and volume move faster

If every length of a figure is multiplied by k, then:

• lengths — perimeter, circumference, a side, a radius — scale by k;
• areas — surface area, cross-section, floor space — scale by k²;
• volumes — capacity, weight of a solid object of the same material — scale by k³.

Double every edge of a box and it holds eight times as much, not twice as much. Triple the radius of a circle and the area is nine times as large. This is also the fastest way to handle maps and scale models: a map scale of 1 : 20,000 is a length scale, so an area on the map represents 20,000² times that area on the ground.

Two more things worth being deliberate about. Order of operations is not a tie-breaker rule invented for tests — it is what the notation means, so 3 + 4 × 5 is 23 and nothing else. And estimation is a scoring strategy, not a compromise: when the choices are far apart, rounding to numbers you can handle in your head gets you there faster and more reliably than exact arithmetic done under time pressure.