SAT Math tutoring online: solve a ratio after one quantity changes

A ratio describes a relationship between quantities, not their actual sizes. If red:blue is 3:5, the numbers could be 3 and 5, 30 and 50, or another common multiple. A later change can provide the information needed to determine the original quantities.

Original teaching problem

A container holds red and blue marbles in a ratio of 3:5. After 12 red marbles are removed and no blue marbles are changed, the ratio of red to blue is 1:3. How many blue marbles are in the container?

This is an original exercise, not an official SAT item. It uses a ratio and a linear equation, skills covered in College Board’s Math content domains.

Name both quantities with one scale factor

Let the original red count be 3x and the blue count be 5x. After the removal, red becomes 3x − 12, while blue stays 5x.

The new ratio gives (3x − 12)/(5x) = 1/3. Multiplying both sides by 15x gives 3(3x − 12) = 5x. Therefore 9x − 36 = 5x, so 4x = 36 and x = 9.

The blue count is 5 × 9 = 45 marbles. The scale factor 9 is an intermediate result, not the answer requested.

Verify the change

The original counts are 27 red and 45 blue, which reduce to 3:5. Removing 12 red leaves 15 red and 45 blue, which reduce to 1:3. The blue count is the same before and after, consistent with the question.

Subtracting 12 directly from the ratio part 3 would mix an actual marble count with an unknown-size part. Removing 12 from both groups would model a different situation.

Recognise what remains constant

On a new problem, write which quantity changes before writing the equation. If a problem adds to one group, changes both groups or specifies a total, the equation must reflect those details. The word “ratio” alone does not prescribe one universal setup.

Use a later mixed practice set to check whether you recognise the structure without a worksheet title announcing it. For a short starting point, try the Math Skills Check. An online SAT lesson can then focus on your attempted setup and the point where the relationship became unclear.