University admissions preparation Math Strategy Without Formulas

Knowing formulas matters, but what SAT Math often tests is this: can you recognize what type of question you are facing? On the digital SAT a calculator is allowed on the entire Math section (Bluebook has a built-in Desmos graphing calculator), yet many questions are best solved step by step with the right strategy, without a calculator. Learn to recognize three situations:

• The answer choices contain specific numbers and the question asks for a value → backsolving: look at the choices not as equations but as candidates to test.
• The choices contain variables, or the question asks for an expression in terms of x → plugging in: substitute simple numbers of your own, and the question turns into ordinary arithmetic.
• The numbers are ugly (0.049, 79.95, a big product) → estimation: round, and eliminate the options whose size doesn't fit.

Multiple-choice questions have just 4 options, so knocking out the wrong ones is as effective as finding the right one. Student-produced response (grid-in) questions have no options, so backsolving doesn't work, but you can always check your answer by substituting it back.

The logic is simple: if the question asks for a number and 4 candidates sit in the options, test the candidates instead of setting up an equation. The steps:

1. Take the options in ascending order (they are usually already sorted).
2. Start in the middle — with B or C.
3. Only when you know the result increases as the tested value increases: if the result is smaller than needed, try a larger option; if it is larger, try a smaller one. Otherwise do not assume a direction; check other candidates separately.
4. If there is more than one condition, the winning option has to pass all of them.

Example. Sara spent half of her money on a book, then paid $8 for lunch and had $12 left. How much money did she start with? A) $32 B) $36 C) $40 D) $48

We start with C ($40): half, that is $20, goes on the book; of the remaining $20, $8 goes on lunch → $12 left ✓. The answer is C. If we had started from B ($36): 18 − 8 = 10 → too little would be left, so we would have had to go up. The classic mistake here creeps in when you try to work the steps ("spent half, then 8") backwards; backsolving needs no reversing at all. Remember: this technique works only on questions with answer choices.

Plug in real numbers

When choices contain variables or the question asks what must be true, testing valid numbers is one possible method; a short algebraic explanation can sometimes be simpler:

• Choose small, simple numbers that fit the conditions (they must meet the positive, integer, odd/even requirements).
• Compute each option with these numbers and cross out those that don't survive.
• If two options survive, test again with a different second number.

Example. If k is any odd integer, which of the following must be an even integer? A) k B) k + 2 C) 2k D) k^2

We take k = 3: A) 3 is odd; B) 5 is odd; C) 6 is even ✓; D) 9 is odd. Only C remains because the other choices fail for this valid input. One example can reject a false option but does not by itself prove a general claim; here twice an integer is always even. Compare accuracy and speed in your own practice.

Eliminate by estimation

Estimation is useful only when it separates the choices reliably. Round, consider the possible error and eliminate distant choices. Calculate more precisely if the choices are close.

Example. A jacket priced at $79.95 is on sale at 24.5% off. Which is closest to the sale price? A) $48 B) $56 C) $60 D) $68

$79.95 ≈ $80, 24.5% ≈ 25%. The discount is about $20, and the amount paid is ≈ $60 → C. A and B are clearly smaller, D is bigger. In this example the estimate separates C from the other choices. An exact check in Desmos is also possible; compare which method is comfortable in practice.