Daily Quant Practice - 1A - Set 1

Q1: For the Multiple Choice Questions, select the correct alternative from the given choices. Solve: $199x+201y=1007$ $201x+199y=999$.

1. $x=2, y=3$
2. $x=3, y=2$
3. $x=3, y=4$
4. $x=4, y=3$

Correct Option: A

Correct Answer:

Explanation:

Q2: For the Multiple Choice Questions, select the correct alternative from the given choices. Solve: $\frac{16}{2x+3y}-\frac{7}{3x+2y}=1$, $\frac{8}{3(2x+3y)}+\frac{21}{3x+2y}=\frac{10}{3}$

1. $x=2, y=1$
2. $x=1, y=2$
3. $x=-2, y=-1$
4. $x=-1, y=-2$

Correct Option: B

Correct Answer: x=1, y=2

Explanation: Step 1: Let <span class="math-inline">a \= \\frac\{1\}\{2x\+3y\}</span> and <span class="math-inline">b \= \\frac\{1\}\{3x\+2y\}</span>. The equations become: (i) <span class="math-inline">16a \- 7b \= 1</span> (ii) <span class="math-inline">\\frac\{8\}\{3\}a \+ 21b \= \\frac\{10\}\{3\}</span> which simplifies to <span class="math-inline">8a \+ 63b \= 10</span>. Step 2: Multiply the simplified (ii) by 2 to get <span class="math-inline">16a \+ 126b \= 20</span>. Step 3: Subtract (i) from this: <span class="math-inline">\(16a\+126b\) \- \(16a\-7b\) \= 20\-1</span>, which gives <span class="math-inline">133b \= 19</span>, so <span class="math-inline">b \= \\frac\{1\}\{7\}</span>. Step 4: Substitute b in (i): <span class="math-inline">16a \- 7\(\\frac\{1\}\{7\}\) \= 1</span>, which gives <span class="math-inline">16a\=2</span>, so <span class="math-inline">a\=\\frac\{1\}\{8\}</span>. Step 5: Substitute back: <span class="math-inline">2x\+3y\=8</span> and <span class="math-inline">3x\+2y\=7</span>. Solving this system gives <span class="math-inline">x\=1, y\=2</span>.

Q3: For the Non-Multiple Choice Questions, write your answer in the box provided. A rope of 77 meters is cut into two pieces such that the length of one piece is $4/7^{th}$ of the other. What is the length of $3/14^{th}$ of the longer piece? (in m)

Correct Option:

Correct Answer: 10.5

Explanation: Step 1: Let the pieces be x and y. <span class="math-inline">x\+y\=77</span> and <span class="math-inline">x \= \\frac\{4\}\{7\}y</span>. Step 2: Substitute x in the first equation: <span class="math-inline">\\frac\{4\}\{7\}y \+ y \= 77 \\implies \\frac\{11\}\{7\}y \= 77 \\implies y \= 49</span>. Step 3: The other piece is <span class="math-inline">x \= 77 \- 49 \= 28</span>. The longer piece is 49 m. Step 4: Calculate <span class="math-inline">3/14^\{th\}</span> of the longer piece: $\frac{3}{14} \times 49 = \frac{3 \times 7}{2} = 10.5 m.

Q4: For the Multiple Choice Questions, select the correct alternative from the given choices. Anand has only 10 paise and 25 paise coins with him. If he has 70 coins in all worth ₹10 with him, how many 25 paise coins does he have?

1. 20
2. 25
3. 40
4. 50

Correct Option: A

Correct Answer: 20

Explanation: Step 1: Let t be the number of 10p coins and q be the number of 25p coins. <span class="math-inline">t\+q\=70</span>. Step 2: The total value is <span class="math-inline">10t \+ 25q \= 1000</span> (since ₹10 = 1000 paise). Step 3: From the first equation, <span class="math-inline">t\=70\-q</span>. Substitute this into the value equation: <span class="math-inline">10\(70\-q\) \+ 25q \= 1000</span>. Step 4: Simplify and solve for q: <span class="math-inline">700 \- 10q \+ 25q \= 1000 \\implies 15q \= 300 \\implies q \= 20</span>. He has 20 coins of 25 paise.

Q5: For the Multiple Choice Questions, select the correct alternative from the given choices. Find the greater of the two numbers such that their sum is 200 and the difference of their squares is 8000.

1. 80
2. 100
3. 120
4. 140

Correct Option: C

Correct Answer: 120

Explanation: Step 1: Let the numbers be x and y. <span class="math-inline">x\+y\=200</span>. Step 2: The difference of their squares is <span class="math-inline">x^2 \- y^2 \= 8000</span>. Step 3: Factor the difference of squares: <span class="math-inline">\(x\-y\)\(x\+y\) \= 8000</span>. Step 4: Substitute the sum: <span class="math-inline">\(x\-y\)\(200\) \= 8000 \\implies x\-y \= 40</span>. Step 5: Solve the system <span class="math-inline">x\+y\=200</span> and <span class="math-inline">x\-y\=40</span>. Adding them gives <span class="math-inline">2x\=240 \\implies x\=120</span>. The other number is <span class="math-inline">y\=80</span>. The greater number is 120.

Q6: For the Multiple Choice Questions, select the correct alternative from the given choices. A fraction becomes $1/2$, if its numerator is increased by 1 and the denominator by 3. It becomes $2/5$ if the numerator is increased by 2 and the denominator by 7. Find the fraction.

1. $1/2$
2. $4/7$
3. $1/5$
4. $2/3$

Correct Option: D

Correct Answer: <span class="math-inline">\\frac\{2\}\{3\}</span>

Explanation: Step 1: Let the fraction be x/y. From the first condition, <span class="math-inline">\\frac\{x\+1\}\{y\+3\} \= \\frac\{1\}\{2\} \\implies 2x\+2 \= y\+3 \\implies 2x\-y\=1</span>. Step 2: From the second condition, <span class="math-inline">\\frac\{x\+2\}\{y\+7\} \= \\frac\{2\}\{5\} \\implies 5x\+10 \= 2y\+14 \\implies 5x\-2y\=4</span>. Step 3: From <span class="math-inline">2x\-y\=1</span>, we get <span class="math-inline">y\=2x\-1</span>. Substitute into the second equation: <span class="math-inline">5x\-2\(2x\-1\)\=4</span>. Step 4: Solve for x: <span class="math-inline">5x\-4x\+2\=4 \\implies x\=2</span>. Then <span class="math-inline">y\=2\(2\)\-1\=3</span>. The fraction is <span class="math-inline">\\frac\{2\}\{3\}</span>.

Q7: For the Multiple Choice Questions, select the correct alternative from the given choices. There is some money with Ajay and some with Vijay. If Ajay gives 30 to Vijay, then the amounts with them would be interchanged. Instead, if Vijay gives ₹20 to Ajay, then Ajay would have ₹70 more than Vijay would have. Find the amount that Ajay has.

1. 40
2. 50
3. 70
4. Cannot be determined

Correct Option: D

Correct Answer:

Explanation:

Q8: For the Non-Multiple Choice Questions, write your answer in the box provided. How many two-digit numbers with their tens digit greater than their units digit, have the sum of their digits equal to twice their difference?

Correct Option:

Correct Answer: 3

Explanation: Step 1: Let the number be 10t+u, with t>u. Step 2: The condition is <span class="math-inline">t\+u \= 2\(t\-u\)</span>. Step 3: Simplify the equation: <span class="math-inline">t\+u \= 2t\-2u \\implies 3u\=t</span>. Step 4: We need to find pairs of digits (t, u) where t=3u and t>u. If <span class="math-inline">u\=1, t\=3</span>. Number: 31. (3>1 OK). If <span class="math-inline">u\=2, t\=6</span>. Number: 62. (6>2 OK). If <span class="math-inline">u\=3, t\=9</span>. Number: 93. (9>3 OK). If <span class="math-inline">u\=4, t\=12</span>. (Not a digit). There are 3 such numbers.

Q9: For the Non-Multiple Choice Questions, write your answer in the box provided. A two-digit number is such that the sum of its digits is thrice the difference of its digits. If the number exceeds the number formed by reversing its digits by 36, find the number.

Correct Option:

Correct Answer: 84

Explanation: Step 1: Let the number be 10t+u. Reversal is 10u+t. Step 2: From the second condition, <span class="math-inline">\(10t\+u\)\-\(10u\+t\)\=36 \\implies 9t\-9u\=36 \\implies t\-u\=4</span>. This means t>u. Step 3: From the first condition, <span class="math-inline">t\+u \= 3\(t\-u\)</span>. Step 4: Substitute <span class="math-inline">t\-u\=4</span> into the first condition: <span class="math-inline">t\+u \= 3\(4\) \\implies t\+u\=12</span>. Step 5: Solve the system <span class="math-inline">t\-u\=4</span> and <span class="math-inline">t\+u\=12</span>. Adding them gives <span class="math-inline">2t\=16 \\implies t\=8</span>. Then <span class="math-inline">u\=4</span>. The number is 84.

Q10: For the Multiple Choice Questions, select the correct alternative from the given choices. The difference between a three-digit number and the number formed by reversing its digits is 297. The sum of the units and the tens digits is the same as the difference of the hundreds and the units digits. Also, the hundreds digit is twice the units digit. Find the number.

1. 242
2. 342
3. 603
4. 884

Correct Option: C

Correct Answer: 603

Explanation: Step 1: Let the number be 100h+10t+u. The difference with its reverse is <span class="math-inline">99\(h\-u\) \= 297 \\implies h\-u\=3</span>. Step 2: The hundreds digit is twice the units digit: <span class="math-inline">h\=2u</span>. Step 3: Substitute h=2u into the first result: <span class="math-inline">2u\-u\=3 \\implies u\=3</span>. Then <span class="math-inline">h\=2\(3\)\=6</span>. Step 4: The sum of units and tens equals the difference of hundreds and units: <span class="math-inline">u\+t \= h\-u</span>. Step 5: Substitute known values: <span class="math-inline">3\+t \= 6\-3 \\implies 3\+t\=3 \\implies t\=0</span>. The number is <span class="math-inline">100\(6\)\+10\(0\)\+3 \= 603</span>.

Q11: For the Non-Multiple Choice Questions, write your answer in the box provided. Four years ago, a man was thrice as old as his son. Eight years hence, the man will be twice as old as his son. What is the present age (in years) of the son?

Correct Option:

Correct Answer: 16

Explanation: Let the present ages of the man and son be M and S. From the given conditions: 1. $M - 4 = 3(S - 4) \implies M - 3S = -8$ 2. $M + 8 = 2(S + 8) \implies M - 2S = 8$ Subtracting the first equation from the second gives S = 16.

Q12: For the Non-Multiple Choice Questions, write your answer in the box provided. The present average age of Ram and his wife Sita and their daughter is 35 years. Fifteen years from now, the age of Sita will be equal to the sum of the present ages of Ram and the daughter. Find the present age (in years) of Sita.

Correct Option:

Correct Answer: 45

Explanation: Let the present ages be R, S, and D. 1. $(R + S + D)/3 = 35 \implies R + S + D = 105$ 2. $S + 15 = R + D Substitute (R + D) from the second equation into the first: $(S + 15) + S = 105 \implies 2S = 90 \implies S = 45$.

Q13: For the Multiple Choice Questions, select the correct alternative from the given choices. X says to Y, "I am twice as old as you were when I was as old as you are." The sum of their present ages is 63 years. Find the present age of X.

1. 24 years
2. 39 years
3. 36 years
4. 42 years

Correct Option: C

Correct Answer: 36 years

Explanation: Let the present ages of X and Y be x and y. 1. $x + y = 63$ 2. The time when X was as old as Y is (x - y) years ago. At that time, Y's age was $y - (x - y) = 2y - x$. The second condition translates to $x = 2(2y - x) \implies 3x = 4y$. Solving the two equations gives $x = 36$ and $y = 27$.

Q14: For the Multiple Choice Questions, select the correct alternative from the given choices. Six years ago, the age of a person was two years more than five times the age of his son. Four years hence, his age will be two years less than three times the age of his son. After how many years from now will their combined age be 100 years?

1. 48 years
2. 14 years
3. 19 years
4. 38 years

Correct Option: C

Correct Answer: 19 years

Explanation: Let the present ages be P and S. 1. $P-6 = 5(S-6)+2 \implies P-5S = -22$ 2. $P+4 = 3(S+4)-2 \implies P-3S = 6$ Solving this system gives S=14 and P=48. Their current combined age is 14+48=62. Let y be the required years. $(48+y) + (14+y) = 100 \implies 62 + 2y = 100 \implies 2y=38 \implies y=19$.

Q15: For the Non-Multiple Choice Questions, write your answer in the box provided. (a) If the following three equations form a system of dependent equations, what is the value of p? I. $3x+2y-7z=56$ II. $5x+3y+z=16$ III. px+12v-19z=200

Correct Option:

Correct Answer: 19

Explanation: For the system to be dependent, one equation must be a linear combination of the others, i.e., Eq. III = a(Eq. I) + b(Eq. II). By comparing the coefficients of y and z: • $2a + 3b = 12$ • $-7a + b = -19$ Solving this gives a=3 and b=2. Now, compare the x-coefficients: p = 3a + 5b = 3(3) + 5(2) = 19.