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Difficulty:
75%
(hard)
Question Stats:
50% (02:27) correct
50%
(02:34)
wrong
based on 30
sessions
History
Date
Time
Result
Not Attempted Yet
If \(1 - \frac{p}{100}\) is used as an approximation to \(\frac{1}{1 + \frac{p}{100}}, \quad 0 < p < 10,\) what is the ratio of the magnitude of the error made to the correct value?
(A) \(\frac{p}{100}\)
(B) \(\left(\frac{p}{100}\right)^2\)
(C) \(\frac{p}{100 - p}\)
(D) \(\frac{p^2}{100}\)
(E) \(\frac{p}{100 + p}\)
(A) \(\frac{p}{100}\)
(B) \(\left(\frac{p}{100}\right)^2\)
(C) \(\frac{p}{100 - p}\)
(D) \(\frac{p^2}{100}\)
(E) \(\frac{p}{100 + p}\)
17 Apr 2026, 06:40
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Step 1: Write the correct value
The correct value is
1 ÷ (1 + p/100)
Multiply numerator and denominator by 100 to simplify:
Correct value = 100 ÷ (100 + p)
Step 2: Write the approximation used
Given approximation is
1 − p/100
Write this as a single fraction:
Approximate value = (100 − p) ÷ 100
Step 3: Find the error made
Error = Approximate value − Correct value
Error = (100 − p)/100 − 100/(100 + p)
Take a common denominator, 100(100 + p):
Error = \[(100 − p)(100 + p) − 10000] ÷ \[100(100 + p)]
Step 4: Simplify the numerator
(100 − p)(100 + p) = 10000 − p2
So numerator becomes:
(10000 − p2) − 10000 = −p2
Magnitude of error = p2 ÷ \[100(100 + p)]
Step 5: Find the ratio of the magnitude of the error to the correct value
Correct value = 100 ÷ (100 + p)
Ratio =
\[p2 ÷ (100(100 + p))] ÷ \[100 ÷ (100 + p)]
Step 6: Simplify the ratio
The term (100 + p) cancels out:
Ratio = p2 ÷ (100 × 100)
Ratio = (p/100)2
Final Answer:
(p / 100)2
Correct option: (B)
The correct value is
1 ÷ (1 + p/100)
Multiply numerator and denominator by 100 to simplify:
Correct value = 100 ÷ (100 + p)
Step 2: Write the approximation used
Given approximation is
1 − p/100
Write this as a single fraction:
Approximate value = (100 − p) ÷ 100
Step 3: Find the error made
Error = Approximate value − Correct value
Error = (100 − p)/100 − 100/(100 + p)
Take a common denominator, 100(100 + p):
Error = \[(100 − p)(100 + p) − 10000] ÷ \[100(100 + p)]
Step 4: Simplify the numerator
(100 − p)(100 + p) = 10000 − p2
So numerator becomes:
(10000 − p2) − 10000 = −p2
Magnitude of error = p2 ÷ \[100(100 + p)]
Step 5: Find the ratio of the magnitude of the error to the correct value
Correct value = 100 ÷ (100 + p)
Ratio =
\[p2 ÷ (100(100 + p))] ÷ \[100 ÷ (100 + p)]
Step 6: Simplify the ratio
The term (100 + p) cancels out:
Ratio = p2 ÷ (100 × 100)
Ratio = (p/100)2
Final Answer:
(p / 100)2
Correct option: (B)
