Question

An exam will have 20 questions worth a total of 100 points. There will be true-false questions worth 3 points each and short answer worth 11 points each. The students want to know how many of each type there will be on the test. The teacher won't tell. How many short answer questions are there?​

Answer 1

4 true false and 8 short answer

Or 15 true false and 5 short answer

Or 26 true false and 2 short answer

Explanation:

let x be the number of true-false questions

y be the number of short-answer questions

Then we have to solve the equation 3x + 11y = 100

Important: x and y are natural numbers

3x + 11y = 100 ⇔ 100 - 11y = 3x

If y=10 ⇒ 100 - 11×10 = −10 = 3x impossible because x ≥ 0

If y=9 ⇒ 100 - 11×9 = 1 = 3x impossible because x is a natural number

If y=8 ⇒ 100 - 11×8 = 12 = 3x ⇒ x = 12/3 = 4

Then (x,y) = (4,8) is a solution to the equation

If y=7 ⇒ 100 - 11×7 = 23 = 3x impossible because x is a natural number

If y=6 ⇒ 100 - 11×6 = 34 = 3x impossible because x is a natural number

If y=5 ⇒ 100 - 11×5 = 45 = 3x ⇒ x = 45/3 = 15

Then (x,y) = (15,5) is a solution to the equation

If y=4 ⇒ 100 - 11×4 = 56 = 3x impossible because x is a natural number

If y=3 ⇒ 100 - 11×3 = 67 = 3x impossible because x is a natural number

If y=2 ⇒ 100 - 11×2 = 78 = 3x ⇒ x = 78/3 = 26

Then (x,y) = (26,2) is a solution to the equation

If y=1 ⇒ 100 - 11×1 = 89 = 3x impossible because x is a natural number

If y=0 ⇒ 100 - 11×0 = 100 = 3x impossible because x is a natural number