In a circle with area of 120 to 124 sq inches, the area of the sector formed by an angle is between 20 and 21 sq inches. What is one possible integer value of the angle?

I came up with a low value of 60 (if area of circle is on the lowest end, 120 sq inches, and area of the sector is also on the low end, 20 inches). If both those areas are on the highest end, then I came up with 60 again. But answer is supposed to be 59 ≤ x ≤ 63. Does this make sense, and if so, can you explain it?

Starting at square 1, the way you get this is to remember that sector areas are proportional to central angles. So you can set up the following general proportion:

All you really need to do here is find one number that works, and 60 does, so you needn’t lose much sleep, but if you want to find the whole range you need to remember what’ll give you the biggest and smallest fractions.

The biggest fraction will come from the biggest numerator and smallest denominator. Biggest numerator here is a sector area of 21 square inches, and smallest denominator is a circle area of 120 square inches.

Then the smallest fraction will be the smallest numerator (20 square inches) over the biggest denominator (124 square inches).

Does that help?

## Comments (1)

Thanks, Mike! My mistake was putting biggest/biggest and smallest/smallest.