dMAT Mathematical Equations: Rules, Mental Method and Practice Questions

By the Yastudy Test Prep Team · Last updated 11 October 2026

Mathematical Equations is the second subtest of the dMAT Core Module. Each task is a small system of equations in which letters stand for whole numbers from 1 to 20. You find the number behind each letter. There are 20 systems in 25 minutes, and you may not write anything down.

How the task works

In the official exercises, a task has two to four short equations with the letters A to D. Your job is to find the one set of numbers that makes every equation true.

The official instructions fix three things:

• Each letter is a whole number between 1 and 20.
• There is always exactly one solution.
• You have no paper and no calculator.

The arithmetic itself is simple. What the subtest measures is whether you can keep a short chain of steps in your head and stay accurate at speed.

A four-step method that works without paper

1. Find the anchor. Look for the equation that gives you a number most directly. Best is an equation with only one letter, such as 5 + A = 12. Next best is one that writes a letter as a multiple of another, such as B = 3 × A.
2. Substitute into the equation with the fewest letters. Replace one letter so that the equation contains a single unknown. B − A = 6 with B = 3 × A becomes 3 × A − A = 6.
3. Solve for that one letter, then say the value to yourself. A = 3. Repeating it fixes it in your memory before you move on.
4. Unfold the rest. Put the value into the other equations one at a time. Each one now gives you a new letter immediately.

If you have a few seconds left, test your numbers in the one equation you did not use for solving. It catches most slips.

Shortcuts for mental work

• Use the 1 to 20 limit. If 5 × B = A, then B can only be 1, 2, 3 or 4. Trying four values is sometimes quicker than algebra.
• Use divisibility. If 5 × C = A and A + C = 18, then A + C is 6 × C, so C is 3. Spotting "six times C" saves a step.
• Collect like terms in words. Think "five B minus B plus ten B is fourteen B" instead of picturing the symbols.
• Handle brackets with care. In A − B + C − D, a letter D that stands for "6 + B" must be subtracted as a whole. That gives minus 6 and minus B.
• Work in one direction. Do not jump between equations. Finish one substitution before you start the next.

Worked example

1. Anchor: the first equation says A is four times C.
2. Substitute into the second: 4 × C + C = 15, so 5 × C = 15 and C = 3.
3. Unfold: A = 4 × 3, so A = 12.
4. Last letter: B = 12 − 2 × 3, so B = 6.

Check with the equation you did not solve with: A + C = 12 + 3 = 15. Correct.

Practice questions

Solve each system in your head, then open the answer. Aim for 75 seconds or less.

Common mistakes

• Starting with the longest equation. The long equation is usually where everything comes together at the end. Start with the short ones.
• Sign errors with brackets. Subtracting a letter that stands for a sum is the classic trap in four-letter systems.
• Losing a value. If you forget a number halfway, do not start again from the top. Re-derive only that letter from its anchor equation.
• Accepting a value above 20 or a fraction. If you get one, you have made an error, because every letter is a whole number from 1 to 20.
• Spending three minutes on one system. Each task counts the same. If you are stuck after about 90 seconds, guess and move on.

What to practise next

• Latin Squares, the third Core subtest
• Figure Sequences, the first Core subtest
• The timed dMAT sample test, which includes ten more systems of equations