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QUESTION: 1

What is the value of x/10 , when it is rounded to the nearest hundredth digit? Take the value of x as 194.283.

Solution:

QUESTION: 2

What is the approximate value of the expression

Solution:

QUESTION: 3

Given p = 5.369 and q = 7.4y5. When q is rounded to the nearest hundredth digit, it is equal to 7.5. Find p + q.

Solution:

QUESTION: 4

What is the value of 47.xy9 when it is rounded to the nearest integer? It is also known that x > y.

- The sum of x and y is 6
- The product of x and y is 5.

Solution:

QUESTION: 5

The expression 32045 × 643 ÷ 15987 is approximately equal to:

Solution:

**Step 1: Question statement and Inferences**

We need to find the approximate value of 32045 × 643 ÷ 15987

As all the answer options have zero in unit and tens digit

We can round all the numbers to hundreds place

**Step 2: Finding required values**

Rounding 32045 to hundreds = 32000

Rounding 643 to hundreds = 600

Rounding 15987 to hundreds = 16000

** **

**Step 3: Calculating the final answer**

32045 × 643 ÷ 15987 ≃ 32000 * 600/16000 = 1200

**Answer: Option (C)**

QUESTION: 6

What is the approximate value of N, if N is equal to 33% of 3.683 × 42.198 ÷ 20.679?

Solution:

**Step 1: Question statement and Inferences**

We have to find approximate value of 33% of 3.683 × 42.198 ÷ 20.679

**Step 2: Finding required values**

As we have to find the approximate value.

Rounding to the units digit:

- 3.683 = 4
- 42.198 = 42
- 20.679 = 21

**Step 3: Calculating the final answer**

Let’s simplify the expression

- N’ = 4 x 42/21
- N’ = 8

The final answer

- N = 0.33 x 8 = 33*8/100 = approx 33*8/99 = approx 8/3 = 2.67

**Answer: Option (C)**

QUESTION: 7

When rounded to the nearest hundredths digit, the number p becomes 6.72. What is the hundredths digit of p?

(1) The thousandths digit of p is 3.

(2) The sum of the hundredths and the thousandths digits of p is 5

Solution:

__Steps 1 & 2: Understand Question and Draw Inferences__

We are given a number p. We have to find the hundredths digit of p.

Let’s say p = a.bcde (Here a, b, c, d and e are digits)

We are given that on rounding to the hundredths digit, p becomes 6.72

So, by comparing the form a.bcde with 6.72, we conclude that:

a = 6, b = 7 and:

If d ≥ 5, then c = 1

If d < 5, then c = 2

We need to find the value of c.

__Step 3: Analyze Statement 1__

Statement 1 says: The thousandths digit of p is 3.

So, the number a.bcde = a.bc3e

That is, d = 3

From the condition above, this clearly implies that c = 2

Thus, Statement 1 alone is sufficient to answer the question: what is the value of c?

__Step 4: Analyze Statement 2__

Statement 2 says: The sum of the hundredths and the thousandths digits of p is 5

That is, c + d = 5

We have inferred that:

If d ≥ 5, then c = 1

In this case, the sum of c + d will range from 6 to 10, inclusive

We also inferred that

If d < 5, then c = 2

This means, if d = 0, 1, 2, 3 or 4, then c = 2

Out of these possible values of d, we see that if d = 3, then c = 2 and the condition given in Statement 2 (that c + d = 5) is also fulfilled.

Therefore, using Statement 2, we can deduce that:

c = 2

Thus, Statement 2 alone is sufficient to find the value of c.

__Step 5: Analyze Both Statements Together (if needed)__

This step is not needed as we get a unique value for c in both steps 3 and 4.

**Answer: Option (D) **

QUESTION: 8

If r, s, and t are the tenths, hundredths, and the thousandths digit respectively of decimal d = 26.rst, is t – 4 positive?

(1) If 10d were rounded to the nearest tenths, the result would be 260.3

(2) s is even

Solution:

**Step 1 & 2 – Understand the question and draw inferences from the question statement. **

Given: d = 26.rst

To find: If t – 4 > 0

- r, s, t are digits
- This means that, r, s are t are integers such that 0 ≤ r, s, t ≤ 9

We need to find if t – 4 > 0

i.e. if t > 4

Thus, we need to find if 5 ≤ t ≤ 9

**Step 3 – Analyze Statement 1 Independently**

Statement 1 – If 10d were rounded to the nearest tenths, the result would be 260.3

d = 26. rst

- 10d = 26r.st

Let’s round 10d to the nearest tenths:

Marking the digit at the tenths place: 26r.**s**t

The digit on **s**’s right is t

i. If t ≥ 5, we will add 1 to **s**.

ii. If t < 5, we will keep **s** as it is

Therefore, 26r.** s**t may be rounded to 26r.**s**0 (if t < 5) or 26r.(**s+1)**0 (if t ≥ 5)

Comparing this with the number 260.3, we get:

- s = 2, if t ≥ 5
- s = 3, if t < 5

Thus, information provided in Statement 1 is not sufficient to arrive at a unique answer.

**Step 4 – Analyze Statement 2 Independently**

Per Statement 2, s is even.

However, to answer the given question, we need to find if 5 ≤ t ≤ 9

Thus, information provided in Statement 2 is not sufficient to arrive at a unique answer.

**Step 5 – Analyze Both Statements Together**

Per Statement 1, we have:

s = 2, if t ≥ 5 and s = 3, if t < 5

Per Statement 2, we have:

s is even.

Combining both statements, we have s = 2

- t ≥ 5

Since t is a digit, t cannot be greater than 9.

- 5 ≤ t ≤ 9

Therefore statement 1 and statement 2 together are sufficient to arrive at a unique answer.

**Answer: Option (C)**

QUESTION: 9

If a = 3.78xy7 and b = 1.37486, where x and y are positive digits, what is the value of a + b?

(1) If a and b are rounded to nearest thousandths digit, then a + b = 5.162

(2) If a and b are rounded to nearest ten-thousandths digit, then a + b = 5.1622

Solution:

__Steps 1 & 2: Understand Question and Draw Inferences__

We are given that:

a = 3.78xy7 (Here, x and y are positive digits)

b = 1.37486

We have to find the value of a + b.

Now, to find the sum of a and b, first we need to find the values of x and y so that we can determine the value of a. Since we already have the value of b, we can then add both of them and find a + b.

So, the question here is what is the value of a?

__Step 3: Analyze Statement 1__

Statement 1 says: If a and b are rounded to nearest thousandths digit, then a + b = 5.162

So, we know that:

(Rounded value of a to nearest thousandths place) + (Rounded value of b to nearest thousandths place) = 5.162 ……………… (1)

Now, we can’t find the rounded value of a, but we can find the rounded value of b to nearest thousandths place.

Now, let’s round b to the nearest thousandths:

- Marking the digit at the thousandths place: 1.37
86__4__ - Since the digit on
**4**’s right is 8 (and therefore, greater than 5), we will add 1 to 4. - Therefore, 1.37486 may be rounded to 1.375

By putting the rounded value of b in equation (1):

(Rounded value of a to nearest thousandths place) + 1.375 = 5.162

(Rounded value of a to nearest thousandths place) = 5.162 – 1.375 = 3.787

So, the rounded value of 3.78xy7 to the nearest thousandths place = 3.787

This means,

If y ≥ 5, then x = 6

If y < 5, then x = 7

Thus, we are not able to determine a unique value of x and y each.

Hence, statement 1 is not sufficient to answer the question: What is the value of a?

__Step 4: Analyze Statement 2__

Statement 2 says: If a and b are rounded to nearest ten-thousandths digit, then a + b = 5.1622

(Rounded value of a to nearest ten-thousandths place) + (Rounded value of b to nearest ten-thousandths place) = 5.1622 ……………… (2)

Let’s round b to the nearest ten-thousandths:

- Marking the digit at the thousandths place: 1.374
6__8__ - Since the digit on
**8**’s right is 6 (and therefore, greater than 5), we will add 1 to 8. - Therefore, 1.37486 may be rounded to 1.3749

By putting the rounded value of b in equation (2):

(Rounded value of a to nearest ten-thousandths place) + 1.3749 = 5.1622

(Rounded value of a to nearest ten-thousandths place) = 5.1622 – 1.3749 = 3.7873

So, the rounded value of 3.78xy7 to the nearest ten-thousandths place = 3.7873

Now, let’s round a to the nearest ten-thousandths:

- Marking the digit at the thousandths place: 3.78x
7__y__ - Since the digit on
**y**’s right is 7, we will add 1 to y. - Now, when we add 1 to y, a becomes 3.7873
- This implies that y + 1 = 3.

So, the value of y = 2.

Thus, a = 3.78x27

Also, by comparing 3.7873 to 3.78x27 we get to know that the value of x is 7.

Thus, we know that the value of a is 3.78727.

So, statement 2 alone is sufficient to answer the question: What is the value of a?

__Step 5: Analyze Both Statements Together (if needed)__

Since statement 2 alone is sufficient to answer the question, we don’t need to perform this step.

**Answer: Option (B) **

QUESTION: 10

j = ab.pqr

Decimal j is represented as above, with digits a, b, p, q and r. If p + q = 3 and q + r = 9, which of the following statements are true?

(1) If j/10 were rounded to the nearest thousandths, the result would contain the digit q.

(2) If 100j were rounded to the nearest tens, the result would contain the digit p.

(3) If 1000j were rounded to the nearest thousands, the result would contain the digit b.

Solution:

Given: j = ab.pqr

- 0 ≤ a, b, p, q, r ≤ 9

Also, p + q = 3

- 0 ≤ p ≤ 3 and 0 ≤ q ≤ 3

((p, q) may be (0,3), (1, 2), (2, 1) or (3, 0))

Also given that q + r = 9

We already concluded from the above that 0 ≤ q ≤ 3,

- 6 ≤ r ≤ 9

((q, r) may be (0, 9), (1, 8), (2, 7) or (3, 6))

Now, let’s analyze the given statements:

**Statement I: If j/10 were rounded to the nearest thousandths, the result would contain the digit q**

j = ab.pqr

- j/10 = a.bpqr

Let’s round j/10 to the nearest thousandths:

- Step R1 -Marking the digit at the thousandths place: a.bp
**q**r - Step R2 -The digit on
**q**’s right is r. - Step R3 – Since r > 5, we will add 1 to
**q**. - Step R4 – a.bp
**q**r will be rounded to a.bp**(q+1)**0.

Conclusion: Statement (1) is FALSE.

**Statement II: If 100j were rounded to the nearest tens, the result would contain the digit p**

j = ab.pqr

- 100j = abpq.r

Let’s round 100j to the nearest tens:

- Step R1 -Marking the digit at the tens place: ab
**p**q.r - Step R2 -The digit on
**p**’s right is q. - Step R3 – Since q < 5, we will keep
**p**as it is. (Note that in fact q < 4.) - Step R4 – ab
**p**q.r will be rounded to ab**p**0.0

Conclusion: Statement (2) is TRUE.

**Statement III: If 1000j were rounded to the nearest thousands, the result would contain the digit b.**

j = ab.pqr

- 1000j = abpqr

Let’s round 1000j to the nearest thousands:

- Step R1 -Marking the digit at the thousands place: a
**b**pqr. - Step R2 -The digit on
**b**’s right is p. - Step R3 – Since p < 5, we will keep
**b**as it is. (Note that in fact p < 4.) - Step R4 – a
**b**pqr will be rounded to a**b**000

Conclusion: Statement (3) is TRUE.

Overall Conclusion: Only statements (2) and (3) are true.

**Answer: Option (D) **

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