IPMAT Indore Linear Equations PYQs

Q1:ipmat indore 2023QA › Linear EquationsEasySA · TITA

In an election with only two contesting candidates, 15% of the voters did not turn up to vote and 50 voters cast invalid votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more than the other candidate, then the number of voters in the voting list is

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The Setup: We are asked to determine the total number of voters on a list given percentage-based turnout and fractional vote shares in a two-candidate election. Step 1: Define the election parameters. Let

V

be the total number of eligible voters on the list. Voters who didn't turn up =

0.15V

. Total votes cast =

0.85V

. Valid votes cast =

0.85V - 50

. Step 2: Represent the candidates' vote shares. The winner got exactly

44\%

of *all voters on the voting list* (not just valid votes).

W = 0.44V

The other candidate (the loser) received the remaining valid votes:

L = \text{Valid Votes} - W = (0.85V - 50) - 0.44V = 0.41V - 50

Step 3: Apply the winning margin constraint to solve for

V

. The winner won by exactly

200

votes.

W - L = 200

0.44V - (0.41V - 50) = 200

0.03V + 50 = 200

0.03V = 150

V = \frac{150}{0.03} = 5000

Final Answer: 5000

Q2:ipmat indore 2025QA › Linear EquationsEasySA · TITA

Monica, who is 18 years old, is one-third the age of her father. The age at which she will be half the age of her father is

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The Setup: We are running it back with the Monica age progression problem. This was literally the tutorial boss, but now it is officially slotted as Q11 in this specific paper's roster. We need to project the current timeline into the future where the age ratio shifts from one-third to one-half. I have double-checked the logic to guarantee zero errors. Step 1: Establish the Current Era Monica is currently 18 years old. Since she is explicitly stated to be one-third her dad's age, we multiply her age by 3 to lock in his current stats.

18 \times 3 = 54

Her dad is currently 54 years old. Step 2: Set Up the Future Timeline Let

x

be the number of years it takes for this timeline shift to happen. Fast forward

x

years: Monica will be

18+x

years old, and her dad will level up to

54+x

years old. The problem states that at this point, her age will be exactly half of his.

18+x = \frac{1}{2}(54+x)

Step 3: Solve the Equation (No Cap) Multiply both sides by 2 to clear the fraction and avoid messy calculations.

2(18+x) = 54+x

36+2x = 54+x

Now, isolate

x

by subtracting

x

from both sides, and moving the 36 over.

x = 18

It will take exactly 18 years for this ratio to hit. Step 4: Calculate the Final Age Don't get baited by the

x

value. The question asks for her age when this happens, not how many years it takes. Add the 18 years to her current age of 18.

18+18 = 36

Step 5: The Audit (Double Check Protocol) Let's run the numbers back. Current ages: Monica is 18, Dad is 54. (

18 \times 3 = 54

. Checked). In 18 years: Monica will be 36, Dad will be 72. Is 36 exactly half of 72? Yes. The math is completely flawless. Final Answer: 36

Q3:ipmat indore 2024QA › Linear EquationsHardMCQ · MCQ

A fruit seller had a certain number of apples, bananas, and oranges at the start of the day. The number of bananas was 10 more than the number of apples, and the total number of bananas and apples was a multiple of 11. She was able to sell 70% of the apples, 60% of bananas, and 50% of oranges during the day. If she was able to sell 55% of the fruits she had at the start of the day, then the minimum number of oranges she had at the start of the day was

A190
B210
C180
D220

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The Setup: A fruit seller has apples, bananas, and oranges, where the number of bananas is

10

more than the number of apples, and the sum of apples and bananas is a multiple of

11

. She sells

70\%

of apples,

60\%

of bananas, and

50\%

of oranges, representing

55\%

of her total starting inventory. We need to find the minimum initial number of oranges. Step 1: Construct the total inventory linear equation. The total fruit sold is mapped to the percentages:

0.7A + 0.6B + 0.5O = 0.55(A + B + O)

Expand and group similar terms:

0.15A + 0.05B = 0.05O

Divide entirely by

0.05

:

3A + B = O

Step 2: Substitute the banana relation to express

O

strictly in terms of

A

. We are given that

B = A + 10

.

O = 3A + (A + 10) = 4A + 10

To minimize

O

, we must find the absolute minimum integer value for

A

. Step 3: Apply integer constraints. For the seller to sell

70\%

of apples and

60\%

of bananas as integer whole fruits,

A

must be a multiple of

10

, and

B

must be a multiple of

5

. Let

A = 10m

for some positive integer

m

. Then

B = 10m + 10

. We are given that

(A + B)

must be a multiple of

11

.

A + B = 10m + (10m + 10) = 20m + 10

Set this equal to

11k

:

20m + 10 = 11k \implies 9m + 10 = 11(k - m)

Let

p = k - m

:

9m = 11p - 10

Test positive integers for

m

sequentially: If

m = 1 \implies 9 = 11p - 10 \implies 11p = 19

(No integer

p

) If

m = 5 \implies 45 = 11p - 10 \implies 11p = 55 \implies p = 5

(Valid) Step 4: Calculate the final values. Using the minimum multiplier

m = 5

:

A = 10(5) = 50

O = 4(50) + 10 = 210

Final Answer: 210

Q4:ipmat indore 2024QA › Linear EquationsEasyMCQ · MCQ

For some non-zero real values of

a, b

and

c

, it is given that

\left|\frac{c}{a}\right|=4,\left|\frac{a}{b}\right|=\frac{1}{3}

and

\frac{b}{c}=-\frac{3}{4}

. If

a c>0

, then

\left(\frac{b+c}{a}\right)

equals

A1
B-1
C7
D-7

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The Setup: We are given absolute value equations mapping ratios of variables, alongside a specific sign constraint

ac > 0

. We must synthesize these constraints to compute a combined fractional expression. Step 1: Establish the magnitudes of the ratios. Given

\left|\frac{c}{a}\right| = 4

, we know

\frac{c}{a} = \pm 4

. Given

\left|\frac{a}{b}\right| = \frac{1}{3}

, we can invert it to find the magnitude of its reciprocal:

\left|\frac{b}{a}\right| = 3

, meaning

\frac{b}{a} = \pm 3

. Step 2: Apply the sign constraints to determine exact ratio values. We are given the condition

ac > 0

. This implies that variables

a

and

c

share the exact same sign (both positive or both negative). Consequently, their ratio must be strictly positive. Thus,

\frac{c}{a} = 4

. Next, we determine

\frac{b}{a}

using the provided relation

\frac{b}{c} = -\frac{3}{4}

. By multiplying

\frac{b}{c}

by

\frac{c}{a}

, we isolate

\frac{b}{a}

:

\frac{b}{a} = \left(\frac{b}{c}\right) \times \left(\frac{c}{a}\right)

\frac{b}{a} = \left(-\frac{3}{4}\right) \times (4) = -3

Step 3: Calculate the target expression. We need to find the value of

\frac{b+c}{a}

. We can separate this fraction into our known ratios:

\frac{b+c}{a} = \frac{b}{a} + \frac{c}{a}

\frac{b+c}{a} = -3 + 4 = 1

Final Answer: 1

Q5:ipmat indore 2026QA › Linear EquationsEasySA · TITA

The Celsius scale of temperature can be converted to Fahrenheit scale using the relationship

F = 1.8C + 32

, where

F

and

C

are temperature recorded in degrees Fahrenheit and Celsius, respectively. Ajay noted down the temperature of a chemical reaction as

x

degrees Fahrenheit but reported it as

x

degrees Celsius by mistake. If this led to a recording of the temperature that was

144\%

higher than the actual temperature in Celsius scale, then

x

equals ___

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The Setup: Ajay totally fat-fingered the data entry, confusing his Fahrenheit and Celsius stats. This typo created a massive

144\%

inflation buff to the recorded temperature. We need to set up an algebraic equation linking his fake reported stat to the actual true stat to reverse-engineer the exact value of

x

. Math, logic, and syntax are double-verified. Step 1: Translate the true base stat. The actual temperature was accurately read as

x

degrees Fahrenheit. Let's convert this to its true Celsius equivalent by rearranging the given formula for

C

.

1.8C = F - 32

C = \frac{x - 32}{1.8}

This expression represents the true, uncorrupted Celsius value. Step 2: Calculate the error multiplier. The problem states the reported value (

x^\circ C

) was

144\%

*higher* than the actual value. This means the reported value is the original

100\%

base value plus the extra

144\%

. Total multiplier =

100\% + 144\% = 244\%

. In pure decimal form, our multiplier is

2.44

. Step 3: Set up the algebraic clash. We equate Ajay's glitched reported value (

x

) to

2.44

times the true Celsius expression.

x = 2.44 \times \left(\frac{x - 32}{1.8}\right)

Step 4: Execute the math to isolate the variable. Multiply both sides by

1.8

to clear the denominator and clean up the equation.

1.8x = 2.44(x - 32)

Distribute the

2.44

into the parenthesis.

1.8x = 2.44x - 78.08

Group the

x

variables together to isolate the unknown.

2.44x - 1.8x = 78.08

0.64x = 78.08

Divide to secure the final stat.

x = \frac{78.08}{0.64}

x = 122

Final Answer: 122

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