francoimps wrote:
BabySmurf wrote:
If \(a = \frac{28}{31}\), \(b =\frac{29}{30}\) and \(c =\frac{31}{33}\). Which of the following is true?
I. \(a+b > 2\)
II. \(b+c-a > 1\)
III. \(a + b^2 - c^2 > 1\)
(A) I only
(B) I and II
(C) II and III
(D) II only
(E) I, II and III
I didn't understand really how to do this in 2-3 minutes. Is there any quick way to solve this or similar problems? Even creating common denominators for this to figure out II. took me a long time. The only statement I could evaluate quickly was I. as all fractions are smaller than 1 and therefore the sum of a+b is obviously not larger than 2. But I struggled with II. and III. Numerators and denominators are all "around 30". So maybe the trick lies in doing something with that information, or is that on purpose as a distraction? Thanks.
Fast Approach (1.16 min):
I. <1 + <1 = <2 hence eliminate
As we eliminated I, we eliminate choices A, B, and E. So we only have to decide between C and D.
C: II and III
D: II only.
Hence, we should only check III. As if III is true, then we pick C. If III is false, then we automatically are left with D.
Check III:
By simple comparison of fractions, you can immediately infer that a < c < b. Estimate:
Assume a = 0.97, c = 0.98 , b= 0.99
III: 0.97 + (0.99)^2 - (0.98)^2 < 1?
even if you do not square 0.99, and 0.98, 0.97+(0.99-0.98) = 0.97 + (.01) < 1. Hence, squaring them would make the number even smaller. So FALSE.
Hence, we are left with choice
D.Hi Francoimps,
Almost everything you have done is correct except one step (Highlighted below) which I am skeptical about being correct all the time
Assume a = 0.97, c = 0.98 , b= 0.99Also one step that you have written
By simple comparison of fractions, you can immediately infer that a < c < b. Estimate:In my opinion this SIMPLE COMPARISON OF FRACTION is not so simple for many and perhaps this consumes some time of the test takers
So I am giving a little clarity here on comparison of such fractions
Given: \(a = \frac{28}{31}\), \(b =\frac{29}{30}\) and \(c =\frac{31}{33}\)
Method 1: Cross multiplicationComparing \(a = \frac{28}{31}\) by writing it on left side and \(b =\frac{29}{30}\) by writing it on Right side
Cross multiply by multiplying the denominators of fractions into the numerator of other fraction to be compared with
i.e. writing on left side \(28*30\) and writing on Right side \(29*31\)
Left side is smaller than right side i.e. a (written on left) will be smaller than b (written on right side)
Similarly compare b with c and know that a < c < b
Method 2: Algebraic Understandinga is smaller than b is very simply visible as the numerator and denominator both as equally less by 1 than numerator and denominator of b
Rule: if u/v<1 and x>0, then u/v < (u+x)/(v+x) Compare \(b =\frac{29}{30}\) and \(c =\frac{31}{33}\)
From b to c, Numerator increased by 2 from 29 to 31, i.e. less than 10% increase
From b to c, Denominator increased by 3 from 30 to 33, i.e. Greater than 10% increase
since numerator has increased by lesser percentage and denominator has increased by greater percentage therefore new fraction (i.e. c) will be smaller than b
Compare \(a = \frac{28}{31}\), and \(c =\frac{31}{33}\)From a to c, Numerator increased by 3 from 28 to 31, i.e. Greater than 10% increase
From a to c, Denominator increased by 2 from 31 to 33, i.e. Less than 10% increase
since numerator has increased by Greater percentage and denominator has increased by Lesser percentage therefore new fraction (i.e. c) will be Greater than a
Hence, a < c < b