OmniTaught / Topics / ACT Math: Deep Dive

AM

Deep dives: section by section

ACT Math: Deep Dive

From linear equations to trig, with backsolving as a weapon.

Start this topic free See all topics

What this covers

Every lesson is short enough to finish in one sitting and ends with questions that make you use what you just read.

  1. Linear Equations and Word-Problem TranslationTurn sentences into equations, then let the choices do the work6 min
  2. Ratios, Percents, and ProportionsOne setup solves them all: part over whole, cross multiply6 min
  3. Functions and QuadraticsPlug in, factor, and read the vertex without breaking stride6 min
  4. Geometry: Angles, Triangles, CirclesA short list of facts covers a long list of questions6 min
  5. Coordinate Geometry and Basic TrigSlope, distance, midpoint, and SOH CAH TOA on a grid6 min
Show all 13 lessonsShow fewer lessons
  1. Statistics, Probability, and CountingAverages you can reverse, odds you can count6 min
  2. Number Sense and SequencesExponents, roots, absolute value, and patterns5 min
  3. Coordinate GeometryLines, distance, midpoint, and circles on the plane5 min
  4. Word Problems and Multi-Step SetupsTranslating language into equations, and answering what was asked5 min
  5. Systems of Equations and InequalitiesTwo unknowns, two equations, and the fastest route between them8 min
  6. The Advanced Topics That Actually AppearLogarithms, complex numbers, matrices, and where they hide8 min
  7. Backsolving, Plugging In, and Strategic GuessingThe multiple-choice weapons that beat doing the algebra8 min
  8. Timed Set: Math Deep20 items, 22 minutes, real pacing22 min

A whole lesson, start to finish

Not a sample and not a summary. This is all of lesson 2, exactly as it reads in the app.

Lesson 2 of 13 · about 6 min

Ratios, Percents, and Proportions

One setup solves them all: part over whole, cross multiply

Three costumes, one machine

Ratios, percents, and proportions look like different topics, but they are the same move wearing different costumes. Each one is a statement that two fractions are equal. Set up the proportion, cross multiply, and solve. On the enhanced ACT this cluster shows up constantly in word problems about prices, maps, recipes, and mixtures.

Key idea

A percent is just a ratio with 100 as the whole: part / whole = percent / 100. If you can write that one line correctly, every percent problem on the test is arithmetic.

Percent change: divide by the original

Percent increase or decrease is always the change divided by the original value, not the new one. A price going from $80 to $92 changed by $12, and 12 divided by 80 is 15 percent. Dividing by 92 instead gives about 13 percent, and that wrong number will be waiting in the choices.

Remember

Discounts run backwards with division. If a shirt sells for $42 after 30 percent off, the original price is 42 / 0.70, because $42 is 70 percent of the original. Adding 30 percent of 42 back on is the trap.

Ratios with real totals

When a ratio like 3:5 comes with a real count, find the value of one ratio unit. If boys to girls is 3:5 and there are 24 boys, one unit is 24 divided by 3, which is 8, so there are 40 girls. This unit method also handles three-part ratios without any new machinery.

Careful

Successive percent changes do not cancel and do not add. Up 10 percent then down 10 percent lands below the start, because the second change is taken from a bigger number. Multiply the factors: 1.10 times 0.90 is 0.99.

Proportion problems reward speed. Map scales, recipe scaling, and unit conversions are all one cross multiplication with a calculator. Write the setup so the same units sit on top of both fractions, and you cannot invert by accident.

Strategy note: percent problems with no starting number given are begging you to pick 100 as the starting value. Every percentage becomes a plain number, and the answer choices check themselves.

Now answer the question that follows it

This is one of 6 questions on that lesson. Pick an answer and it will tell you.

The ratio of boys to girls in a school is 3:5. If there are 24 boys, how many girls are there?

Terms you will own

Cards come back on a schedule built from how well you knew them last time, so the ones you keep missing show up more.

More in deep dives: section by section

AEACT English: Deep DiveEvery grammar rule and rhetoric move the test can throw ARACT Reading: Deep DiveProve every answer from the passage, at speed ASACT Science: Deep DiveA reading test in a lab coat: graphs, trends, and dueling scientists

Start ACT Math: Deep Dive free

Every topic on the site is open during the beta. Nothing to install, and your progress follows you between devices.

Start learning free