Data Sufficiency

BTGmoderatorDC wrote:When the positive number a is rounded to the nearest tenth, the result is the number b. What is the tenths digit of a?

(1) When a is rounded to the nearest integer, the result is less than a.

(2) When b is rounded to the nearest integer, the result is greater than b.

OA C

Source: Manhattan Prep

Say,

a = x.yz, where x is units' digit, y is tenth digit, and z is hundredth digit and
b = x.p, where x is units' digit, and p is tenth digit (p is either y or y + 1)

We have to get the value of y.

Let's take each statement one by one.

(1) When a is rounded to the nearest integer, the result is less than a.

Say the result is a' such that a' < a

Or, a' = x, we see that x < x.yz

Here, y can be any digit 0, 1, 2, 3, or 4. No unique value of y. Insufficient.

(2) When b is rounded to the nearest integer, the result is greater than b.

Say the result is b' such that b' > b

Or, b' = x + 1, we see that (x + 1) > x.p

Here, p can be any digit 5, 6, 7, 8, or 9. The value of p will depend on the values of y and z.

Case 1: Say a = x.46, then b = x.5 and b' = x + 1. Here, y = 4
Case 2: Say a = x.56, then b = x.6 and b' = x + 1. Here, y = 5

No unique value of y. Insufficient.

(1) and (2) together

Considering Statement 1, we cannot have Case 2 as a valid case, else for a = x.56, we will have a' = x + 1 > a (= x.56); this is invalid since as per Statement 1, we must have a' < a.

This implies that a must be x.4z, where z ≥ 5

This way, we have

• a = x.4z
  a' = x < a (= x.4z)
  b = x.5, since z ≥ 5 and
  b' = x + 1 > b (= x.5)

Thus, the tenths digit of a = 4. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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