If 0 < a < b < c, Which of the Following Statements Must be True? GMAT Problem Solving

Question: If 0 < a < b < c, which of the following statements must be true?

1. 2a > b+c
2. c-a > b-a
3. \(\frac{c}{a}<\frac{b}{a}\)

A) I only
B) II only
C) III only
D) I and II
E) II and III

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
Case I. 2a > b + c
Consider this scenario: a = 1, b = 2 and c = 3. This satisfies the given condition that 0 < a < b < c.
HOWEVER, if we plug these values into a statement I, we see that it is NOT the case that 2a > b + c
So, the statement I is Not True.

Case II. c – a > b - a
It's already given that c > b
If we subtract ANY VALUE (such as a) from both sides, the inequality remains valid.
So, statement II is True.

Case III. c/a < b/a
Consider this scenario: a = 1, b = 2 and c = 3. This meets the given condition that 0 < a < b < c.
HOWEVER, if we plug these values into statement III, we see that it is NOT the case that c/a < b/a
So, statement III is Not True.

Hence, II only (option B) is correct.

Approach Solution 2:
Following the rule: 0<a<b<c Given, a=2; b=3; c=4

Putting the above-given conditions:

1. 2a > b + c = 2*2 > 3+4 = 4 > 7(false)
2. c-a >b-a = 4-2 >3-2 = 2 > 1(true)
3. \(\frac{c}{a}<\frac{b}{a}=\frac{4}{2}<\frac{3}{2}=2<1.5\)(false)

Hence, II only (option B) is correct.

Approach Solution 3:
As given in the problem statement: 0 < a < b < c, it is required to find which of the cases are true.

I. 2a > b + c
Plugging numbers a = 1, b = 2 and c = 3, we get:
2*1 > 2 + 3
2 > 5 -- irrelevant

Hence, this case is not true

II. c – a > b - a
In the given equation: 0 < a < b < c, by subtracting a from each we get:
=> 0 - a < a - a < b - a < c - a
Hence c - a > b - a is True

III.  \(\frac{c}{a}<\frac{b}{a}\)

Since b < c as given in the question
Dividing by 'a' which is positive does not impact inequality
Hence \(\frac{b}{a}<\frac{c}{a}\)
Therefore, this case is also not true.

Hence, II only (option B) is correct.

“If 0 < a < b < c, which of the following statements must be true”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide”. To solve the GMAT Problem Solving questions, the candidates must have a basic understanding of mathematics and calculations. The candidates can explore varieties of questions from the GMAT Quant practice papers that will help them to improve their mathematical knowledge.

Suggested GMAT Problem Solving Questions

• A 100-litre Mixture of Milk and Water Contains 36 litres of Milk GMAT Problem Solving
• The Figure shows the Graph of y = (x + 1)(x - 1)^2 in the xy-plane GMAT Problem Solving
• On the Graph above, When x = 1/2, y = 2; and when x = 1, y = 1 GMAT Problem Solving
• A Box Contains 100 Balls, Numbered From 1 To 100 GMAT Problem Solving
• A Hall Is 15 M Long And 12 M Broad GMAT Problem Solving
• The Speed Of A Car Increases By 2 Kms After Every One Hour GMAT Problem Solving
• The Vertices Of A Rectangle In The Standard (X,Y) Coordinate Place Are GMAT Problem Solving
• For Any Positive Integer N, The Sum Of The First N Positive Integers GMAT Problem Solving
• What is the Sum of the First 100 Positive Odd Numbers? GMAT Problem Solving
• A Rectangle is Inscribed in a Circle of Radius r. If the Rectangle is GMAT Problem Solving
• At 3:30 P.M. the Angle Between the Hands of a Clock is GMAT Problem Solving
• For Any Positive Integer n, the Length of n is Defined as GMAT Problem Solving
• A Certain Drive-in Movie Theater has Total of 17 Rows of Parking Spaces GMAT Problem Solving
• If 1 kilometer is Approximately 0.6 Mile, Which of the GMAT Problem Solving
• A Vending Machine is Designed to Dispense 8 Ounces of Coffee GMAT Problem Solving
• A Certain Experimental Mathematics Program was Tried Out in 2 Classes GMAT Problem Solving
• The Ages of the Two Persons Differ by 20 Years. If 5 Year Ago, The Older GMAT Problem Solving
• Tanya Prepared 4 Different Letters To Be Sent To 4 Different Addresses GMAT Problem Solving
• How Many Liters Of Pure Alcohol Must Be Added To A 90-Liter Solution GMAT Problem Solving
• What Is The Remainder When 25^25 Is Divisible By 26 GMAT Problem Solving
• A Quality Control Analyst Collected 200 Measurements Of Length GMAT Problem Solving
• 1/2 + 1/3 + 1/4 + 1/5 + 1/6 GMAT Problem Solving
• A Regular Pentagon is Inscribed in a Circle. If A and B are GMAT Problem Solving
• A Certain Culture of Bacteria Quadruples Every Hour GMAT Problem Solving
• If 2^x + 2^y = x^2 + y^2, Where x and y are Nonnegative GMAT Problem Solving
• If 3 < x < 100, for how many values of x is x/3 the square GMAT Problem Solving
• Hot Competition 3 Sep/8AM: In A Triangle ABC, Shown Above, Three Lines GMAT Problem Solving
• A Thief Steals a Car at 2.30 p.m. and Drives it at 60 km/h. The Theft GMAT Problem Solving
• If |x + 1 | > 2x - 1, Which Of The Following Represents The Correct Range GMAT Problem Solving
• Which Of The Following Is A True Statement? GMAT Problem Solving
• A machine puts c caps on bottles in m minutes. How many hours will it GMAT Problem Solving
• What is the least common multiple of 15, 18, and 24? GMAT Problem Solving
• The Price of Darjeeling Tea (in rupees per kilogram) is 100+0.10n GMAT Problem Solving
• A 5-meter-long wire is cut into two pieces. If the longer piece GMAT Problem Solving
• The average weight of 15 items is 8 kg. The least average GMAT Problem Solving
• In a class of 78 students 41 are taking French, 22 are taking German GMAT Problem Solving
• If |x| > 3, which if the following must be true? GMAT Problem Solving
• 100 students appeared for two examinations. 60 passed the first GMAT Problem Solving
• An integer between 1 and 300, inclusive, is chosen at random GMAT Problem Solving
• In How Many Ways Can 6 Chocolates Be Distributed Among 3 Children GMAT Problem Solving
• A Man Can Row 50 Km Upstream And 72 Km Downstream In 9 Hours GMAT Problem Solving
• A Sum Of Money Doubles Itself In 7 Years. In How Many Years It Becomes Four Fold GMAT Problem Solving
• On Selling A Pen At 5% Loss And A Book At 15% Gain, Karim Gains Rs. 7 GMAT Problem Solving
• If Two Typists Can Type Two Pages in Two Minutes, How Many GMAT Problem Solving
• An Empty Pool being Filled with Water at a Constant Rate takes 8 hours GMAT Problem Solving
• For Every Positive Even Integer n, the Function h(n) is Defined to be GMAT Problem Solving
• The formula F = 9/5 * C + 32 gives the relationship between the temperature GMAT Problem Solving
• A Takes 5 More Days Than B To Do A Certain Job And 9 Days More Than C GMAT Problem Solving
• A Watch Which Gains 5 Seconds In 3 Minutes Was Set Right At 7 A.M. GMAT Problem Solving
• Of The 800 Employees Of Company X, 70 Percent Have Been With The Company GMAT Problem Solving