Simplification Test - 9

1/7 = 0.1429
2/9 = 0.2222
4/11 = 0.3636
3/10 = 0.3
So, the greatest fraction is 4/11

Let the two-digit number be 10x + y, where x is the tens digit and y is the units digit.

1. The sum of the digits is 9:
   x + y = 9

2. When the digits are reversed, the new number is 10y + x. The number decreases by 45 when reversed:
   10x + y - (10y + x) = 45
   Simplify:
   9x - 9y = 45
   x - y = 5

3. Solve the two equations:
   x + y = 9
   x - y = 5

   Add the two equations:
   2x = 14
   x = 7

   Substitute x = 7 into x + y = 9:
   7 + y = 9
   y = 2

4. The number is:
   10x + y = 10(7) + 2 = 72

Answer: D: 72

Let the two-digit number be represented as 10a + b, where:

• a is the tens digit.
• b is the one digit.

According to the problem:

1. The sum of the digits is 8:

   a+b=8

2. When the digits are reversed, the number is decreased by 54. The number with reversed digits is 10b + a. So:

   10a+b −(10b+a)=54

   Simplifying this equation:

   10a+b −10b −a=54
   9a −9b=54
   a −b=6

Now, we have two equations:

1. a+b=8
2. a −b=6

Add these two equations:

(a+b)+(a −b)=8+6
2a = 14
a=7

Substitute a=7 into a+b=8:

7+b=8 , b = 1

So, the number is 10a+b=10 ×7+1=71

Answer: The number is 71.

Let the number be x.
x + x² = 156
x²   + x −156 = 0
(x −12)(x + 13) = 0

x + 13 = 0
x = −13  (Since x is a natural number)
x −12 = 0
x = 12

To determine the unit place of (5427)⁶⁴¹ , consider:

• The unit digit of 5427 is 7 .
• Focus on the pattern of powers of 7.

Powers of 7 cycle through a pattern in their unit digits:

• 7¹ = 7 (unit digit is 7)
• 7² = 49 (unit digit is 9)
• 7³ = 343 (unit digit is 3)
• 7⁴ = 2401 (unit digit is 1)

This pattern (7, 9, 3, 1) repeats every 4 powers. To find the unit digit of (5427)⁶⁴¹ :

• Compute 641 mod 4 to find the position in the cycle.
• 641 mod 4 = 1 , indicating the unit digit corresponds to 7¹ .

Therefore, the unit digit of (5427)⁶⁴¹ is 7 .

The correct answer is B as the unit place will be 7 only .

In these type of of questions we apply a trick by just dividing the given remainder by given no.
60 ÷28
remainder will come( 4)

Let the two numbers be x and y.

1. Given x ²+ y ²= 85 and x ²= y ²+ 77.

2. Substituting x ²= y ²+ 77 into x ²+ y ²= 85:
   (y ²+ 77) + y ²= 85
   2y ²= 8
   y ²= 4, so y = ±2.

3. Substituting y ²= 4 into x ²= y ²+ 77:
   x ²= 4 + 77 = 81, so x = ±9.

4. The product is x ×y = 9 ×2 = 18.

Answer: A: 18

321 raised to 321=1 in unit digit and any number  ending in 5 raised to any number  has always 5 as unit digit, so 1 ×5=5

The given expression is:
(4.669 ×4.669 - 2.331 ×2.331) / ((4.669)²+ (2.331)²- 4.669 ×4.662)

1. Numerator:
   4.669 ×4.669 = 21.794
   2.331 ×2.331 = 5.433
   21.794 - 5.433 = 16.361

2. Denominator:
   (4.669)²= 21.794
   (2.331)²= 5.433
   4.669 ×4.662 = 21.748
   21.794 + 5.433 - 21.748 = 5.479

3. Final Calculation:
   16.361 / 5.479 ≈2.99

Answer: B: 2.99

1. Solve inside the parentheses first:
   7 ÷7 = 1
   So, the expression becomes:
   [7 + 7 ×(7 + 1)] + 7 ÷7

2. Simplify the part inside the parentheses:
   7 + 1 = 8
   Now, the expression becomes:
   [7 + 7 ×8] + 7 ÷7

3. Next, perform the multiplication:
   7 ×8 = 56
   So the expression becomes:
   [7 + 56] + 7 ÷7

4. Add the numbers inside the brackets:
   7 + 56 = 63
   So, the expression becomes:
   63 + 7 ÷7

5. Finally, divide:
   7 ÷7 = 1
   Now, the expression becomes:
   63 + 1 = 64