Two temperature scales, A and B, are related linearly. A reading of 14 degrees…
2023
Two temperature scales, A and B, are related linearly. A reading of 14 degrees on scale A is equivalent to 36 degrees on scale B, and a reading of 133 degrees on scale A is equivalent to 87 degrees on scale B. At what reading will both scales show the same value?
Answer: B. 52.5 — Concept: When two temperature scales are linearly related, every reading on one scale maps to the other through a straight-line relation B = mA + c, where m…
- A.
25.25
- B.
52.5
- C.
75.25
- D.
100.25
Show answer & explanation
Correct answer: B
Concept: When two temperature scales are linearly related, every reading on one scale maps to the other through a straight-line relation B = mA + c, where m is the slope (rate of change of B with respect to A) and c is the intercept. Given any two corresponding pairs of readings, m and c can be found by solving simultaneous equations; the coincidence point is the value of A for which A = B under that relation.
Application:
Let the linear relation between the scales be B = mA + c.
Using the pair (A = 14, B = 36): 36 = 14m + c.
Using the pair (A = 133, B = 87): 87 = 133m + c.
Subtracting the first equation from the second: 87 - 36 = (133 - 14)m, so 51 = 119m, giving m = 51/119 = 3/7.
Substituting m = 3/7 into 36 = 14m + c: 36 = 14(3/7) + c = 6 + c, so c = 30.
The relation between the scales is B = (3/7)A + 30.
For both scales to show the same reading, set A = B: A = (3/7)A + 30.
Rearranging: A - (3/7)A = 30, i.e. (4/7)A = 30, so A = 30 x 7/4 = 52.5.
Cross-check: Substitute A = 52.5 back into the relation: B = (3/7)(52.5) + 30 = 22.5 + 30 = 52.5, confirming A = B = 52.5. The same relation also correctly reproduces the given data at A = 133: B = (3/7)(133) + 30 = 57 + 30 = 87, matching the stem, so the relation itself is verified before being used to find the coincidence point.