Inequalities

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BTGmoderatorRO: Moderator; Posts: 772; Joined: Wed Aug 30, 2017 6:29 pmFollowed by: 6 members

by BTGmoderatorRO » Sat Oct 28, 2017 2:45 am

If a is less than b, which of the following numbers is greater than a and less than b ?

(A) (a + b)/2
(B) ab/2
(C) b^2 - a^2
(D) ab
(E) b - a

which of the above options is the correct answer based on your calculation?
i don't seem to get any right

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Source: Beat The GMAT — Problem Solving | Sat Oct 28, 2017 2:45 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Oct 28, 2017 3:18 am

The question stem should read as follows:

If a is less than b, which of the following expressions MUST BE greater than a but less than b?

(A) (a + b)/2
(B) ab/2
(C) b^2 - a^2
(D) ab
(E) b - a

Test a=0 and b=1 in the answer choices:

(A) (a + b)/2 --> (0+1)/2 = 1/2.
(B) ab/2 --> (0*1)/2 = 0.
(C) bÂ² - aÂ² --> 1Â² - 0Â² = 1.
(D) ab --> 0*1 = 0.
(E) b - a --> 1-0 = 1.

Eliminate B, C, D and E, which do not yield values between a=0 and b=1,

The correct answer is A.

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Sat Oct 28, 2017 3:24 am

Hello Roland. I hope you're well.

You can pick two numbers and try option by option. Let's take a=1 and b=2. Now,

$$Option\ E:\ b-a=2-1=1,$$ this is not greater than a. INCORRECT.

$$Option\ D:\ ab=1\cdot2=2,$$ this is not less than b. INCORRECT.

$$Option\ C:\ b^2-a^2=2^2-1^2=4-1=3,$$ this is not less than b. INCORRECT.

$$Option\ B:\ ab/2=1\cdot2/2=1,$$ this is not greater than a. INCORRECT.

$$Option\ A:\ (a+b)/2=(1+2)/2=3/2=1,5,$$ this is greater than a and less than b. CORRECT.

So, the correct option is A.

I hope this can help you.

I'm available if you'd like any follow up.

Regards.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Tue Nov 12, 2019 7:19 pm

BTGmoderatorRO wrote:If a is less than b, which of the following numbers is greater than a and less than b ?

(A) (a + b)/2
(B) ab/2
(C) b^2 - a^2
(D) ab
(E) b - a

which of the above options is the correct answer based on your calculation?
i don't seem to get any right

Let's let a = 3 and b = 4 and test each answer choice:

A) (a + b)/2 = (3 + 4)/2 = 3.5

We see that (a + b)/2 is greater than a and less than b.

We could stop here, but let's double-check the remaining answer choices.

B) ab/2 = 3(4)/2 = 6

We see that ab/2 is greater than b.

C) b^2 - a^2 = 16 - 9 = 7

We see that b^2 - a^2 is greater than b.

D) ab = 3(4) = 12

We see that ab is greater than b.

E) b - a = 4 - 3 = 1

We see that b - a is less than a.

The only answer choice that produced a value between a and b is A.

Alternate solution:

We are looking for a number that is between the two numbers a and b. We see that (a + b)/2 is the average of the two numbers and the average of any two numbers is always between the two numbers.

Answer: A

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Wed Nov 13, 2019 7:01 am

BTGmoderatorRO wrote:If a is less than b, which of the following numbers is greater than a and less than b ?

(A) (a + b)/2
(B) ab/2
(C) b^2 - a^2
(D) ab
(E) b - a

If a and b are DIFFERENT numbers, then the average of a and b will always be BETWEEN a and b
Since we're told a < b, we know a and b are DIFFERENT.
So, their average, (a +b)/2 must be BETWEEN a and b

Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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