Practice Questions: Simplification

(a) 120
(b) 150
(c) 110
(d) 130
(e) 140

Ans: (b)
Solution: 

The Fast-Track Solution Break the equation into three main components and solve them mentally:

1. Squares and Cubes:
   • $(18)^2 = 324$
   • $(4)^3 = 64$
   • Sum: $324 + 64 = \mathbf{388}$
2. Percentage Calculation:
   • $150\% \text{ of } 110$ is just $110 + 55$ (which is half of 110).
   • Result: $\mathbf{165}$
3. Combine and Solve for ?:
   • $(388 - 165) + ? = 373$
   • $223 + ? = 373$
   • $? = 373 - 223$
   • $? = 150$

Correct Option: (b). 150

Exam Pro-Tips for Speed

• Unit Digit Check: If you are in a rush, look at the last digits: $4 + 4 - 0 + ? = 3$. This gives $8 + ? = 3$. The only way to get a last digit of 3 is if $?$ ends in 0 (e.g., $8 + 5 = 13$

• The "150%" Trick: Always treat $150\%$ as $1.5$ or $1 + \frac{1}{2}$. It is much faster to add half of the number back to itself than to multiply by 1.5.

• Memorization: For banking exams, ensure you have squares memorized up to 30 and cubes up to 15. It saves the 5-10 seconds that usually make the difference.

(a) 72
(b) 56
(c) 92
(d) 96
(e) 48

Ans: (c)
Solution: 

The Fast-Track Solution

1. Simplify the Multiplication/Division Block:

Don't do $16 \times 15$ first. Write it like this in your head:

• Divide $15$ and $20$ by $5$: You get $\frac{16 \times 3}{4} \times 48$.
• Divide $16$ by $4$: You get $4 \times 3 \times 48$.
• Now you have: $12 \times 48$.

2. Multiply $12 \times 48$ (The "Double & Half" or Distribution Trick):

• $12 \times 50 = 600$
• $12 \times 2 = 24$
• $600 - 24 = \mathbf{576}$

3. Subtract the Square:

• $(22)^2 = \mathbf{484}$ (This should be memorized for speed).
• $576 - 484 = \mathbf{92}$

Correct Option: (c). 92

Banker's Shortcut: The Last Digit Method

If you are down to the last 10 seconds:

1. Term 1: $(6 \times 5) \div 0$ is tricky, so look at $\frac{16 \times 15}{20} \times 48 \rightarrow \frac{...0}{20} \times 8 \rightarrow \frac{240}{20} \times 48 = 12 \times 48$. The last digit is 6 ($2 \times 8 = 16$).

2. Term 2: $22^2$ ends in 4 ($2 \times 2 = 4$).

3. Final Digit: $6 - 4 = \mathbf{2}$.

4. Looking at the options, only 92 and 72 end in 2. Since $12 \times 48$ is roughly $12 \times 50 = 600$, and $600 - 484$ is clearly around $100$, 92 is the only logical choice.

Q3. $?^2 + \{20 + (98 \div 14)\} = 91$
(a) 9
(b) 6
(c) 8
(d) 7
(e) 5

Ans: (c)
Solution: 

The Fast-Track Solution

1. Tackle the Parentheses (Mental Math):

• $98 \div 14$
• Trick: If you don't know the table of 14, look at the units digit. $4 \times ? = \text{ends in } 8$. It's either $4 \times 2$ or $4 \times 7$. Since $14 \times 2$ is too small, try $14 \times 7$.
• $10 \times 7 = 70$, $4 \times 7 = 28 \rightarrow 70 + 28 = 98$.
• Result: 7

2. Solve the Brackets: $20 + 7 = \mathbf{27}$

3. Isolate the Square:

• $?^2 + 27 = 91$
• $?^2 = 91 - 27$
• Subtraction Trick:$91 - 20 = 71$, then $71 - 7 = \mathbf{64}$.

4. Final Step:

• $?^2 = 64$
• $\mathbf{?} = 8$

Correct Option: (c). 8

Exam Pro-Tips for Speed

• Tables up to 20: Knowing that $14 \times 7 = 98$
• Unit Digit Shortcut: Look at the equation as $?^2 + (0 + 7) = 1$. This means $?^2$ must end in a 4 ($11 - 7 = 4$). Look at your options:
  • $9^2 = 81$ (ends in 1)
  • $6^2 = 36$ (ends in 6)
  • $8^2 = 64$ (ends in 4)$\rightarrow$Winner!
  • $7^2 = 49$ (ends in 9)

Q4.  $208 \div 52 \times ? \div 64 = 28$
(a) 345
(b) 456
(c) 448
(d) 468
(e) 480

Ans: (c)
Solution:

The Fast-Track Solution

1. Simplify the first division (Mental Math):

• $208 \div 52$
• The Trick: Look at the relationship. $50 \times 4 = 200$ and $2 \times 4 = 8$.
• So, $208 \div 52 = \mathbf{4}$.

2. Rewrite the equation to isolate the unknown: $4 \times \frac{?}{64} = 28$

3. Cancel out the terms:

• Divide both sides by 4: $\frac{?}{64} = 7$
• Now, $? = 64 \times 7$

4. Final Multiplication (The Break-up Trick):

• $60 \times 7 = 420$
• $4 \times 7 = 28$
• $420 + 28 = \mathbf{448}$

Correct Option: (c). 448

Speed Hack: Unit Digit & Estimation

• Unit Digit: From the step $64 \times 7$, calculate the last digit only: $4 \times 7 = 2\mathbf{8}$
• Look at the options: Only C (448) and D (468) end in 8.
• Estimation:$60 \times 7$ is $420$. Since we are multiplying $64$, the answer must be slightly higher than $420$. 448 fits perfectly, while $468$ is too far away.

Ans: (d)
Solution: 

The Fast-Track Solution

Instead of multiplying $68 \times 42$, look for a common number you can pull out to simplify the equation.

1. Spot the Factor of 14:

• Notice that $42$ is $14 \times 3$.
• The equation becomes: $68 \times (14 \times 3) - 14 = ? \times 203$
• Pull out 14 from the left side: $14 \times (68 \times 3 - 1)$

2. Simplify the Brackets:

• $68 \times 3$: Think $(70 \times 3) - (2 \times 3) = 210 - 6 = 204$.
• Now subtract the 1: $204 - 1 = \mathbf{203}$.

3. The "Aha!" Moment:

• Your equation is now: $14 \times 203 = ? \times 203$
• Cancel 203 from both sides.
• $? = 14$

Correct Option: (d). 14

Speed Hack: The Unit Digit Check

If you didn't see the factor of 14, you could use the last digit:

1. Left Side: $(8 \times 2) - 4 \rightarrow 16 - 4 = 12$. The last digit is 2.

2. Right Side: $? \times 3$ must end in 2.

3. Check Options:

   • $21 \times 3 = ...3$

   • $20 \times 3 = ...0$

   • $28 \times 3 = ...4$

   • $14 \times 3 = ...12$ (Ends in 2!) $\rightarrow$ Matches.

   • $24 \times 3 = ...12$ (Also ends in 2, but 14 is a more likely factor given the 203).

By spotting the 14, you solved a "hard" multiplication problem in about 10 seconds without actually doing any long multiplication!