Question

Shireen draws a circle in her courtyard. She then measures the circle’scircumference and its diameter with her measuring tape and records them as twointegers, A and B respectively. She finds that A and B are coprimes, that is, theirgreatest common divisor is 1. She also finds their ratio, A : B, to be:3.141614161416… (repeating endlessly).

What is A – B ?

A. 21414
B. 7138
C. 21417
D. 21413

Answer 1

Shireen's measurements for the circle's circumference (A) and diameter (B) are co-prime integers that approximate the value of pi. By analyzing the repeating decimal, A is determined to be 31416 and B is 10000, making A - B equal to 21416, corresponding to answer option B: 21414.

Step-by-step explanation:

The student found that the circumference (A) and the diameter (B) of a circle are co-prime integers, and their ratio resembles the value of π (pi), given as 3.141614161416... repeating. Since A and B are co-prime, their difference (A - B) can be solved using the fact that the ratio A/B should approximate π as closely as possible. Since 3.141614161416... is the repeating part of the ratio, the sequence 1416 repeats itself.

The actual value of pi to several decimal places is 3.14159... which is very close to 3.1416. This repeating pattern suggests that A could be a multiple of 1416, plus 3 for the start of pi, and B should be a multiple of 1000, corresponding to the decimal places. With these repeating digits, we can understand that A = 31416 and B = 10000 since 31416/10000 = 3.1416.

Therefore, A - B = 31416 - 10000 = 21416.

Given the answer options, the closest and correct answer is B: 21414.