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Ratio, Proportion, Variation and Partnership questions for IPMAT, with solutions

10 free Ratio, Proportion, Variation and Partnership practice questions for IPMAT Indore, Rohtak and JIPMAT, 2 from past papers, each with the answer and a full worked solution. No sign-in. Try each one first, then open the solution. The lessons for this chapter are in Ratio, Proportion, Variation and Partnership on Learn.

1. Question 1Multiple choiceWritten by IPMATwayProportion and Variation

   If $\frac ab=\frac cd=\frac ef=\frac34$, then $\frac{2a+3c-e}{2b+3d-f}$ equals

   1. A.$\frac34$
   2. B.$\frac{9}{4}$
   3. C.$1$
   4. D.$\frac43$
   Show the answer and solution

   Answer: A. $\frac34$

   If several equal ratios equal $k$, any combination $\frac{pa+qc+re}{pb+qd+rf}$ (same multipliers) also equals $k$, as long as the denominator is non-zero. Here $k=\frac34$.

2. Question 2Type the answerWritten by IPMATwayProblems on Ages

   The ratio of the present ages of P and Q is $5:3$. Twelve years ago, P was three times as old as Q. What is the sum of their present ages (in years)?

   Show the answer and solution

   Answer: 48

   Let the present ages be $5k$ and $3k$. Twelve years ago they were $5k-12$ and $3k-12$. Then P was three times Q: $5k-12=3(3k-12)=9k-36$. So $4k=24$ and $k=6$. The ages are 30 and 18, and their sum is $48$.

3. Question 3Type the answerWritten by IPMATwayProblems on Ages

   The average age of a husband and wife was $24$ years when they married. Six years later, the average age of the husband, the wife and their child is $21$ years. How old is the child?

   Show the answer and solution

   Answer: 3

   At the wedding the couple's average was 24, so their ages added to $2\times24=48$. Six years later each is 6 years older, so together they are $48+12=60$. The family of three now totals $3\times21=63$. So the child is $63-60=3$ years old.

4. Question 4Type the answerWritten by IPMATwayProblems on Ages

   A mother is $27$ years older than her daughter. In $5$ years the mother's age will be $4$ years more than twice the daughter's age. What is the mother's present age?

   Show the answer and solution

   Answer: 45

   Let the daughter be $D$ now, so the mother is $D+27$. In 5 years they will be $D+5$ and $D+32$. Then $D+32=2(D+5)+4=2D+14$. So $D=18$. The mother is $18+27=45$.

5. Question 5Type the answerWritten by IPMATwayPartnership

   Three farmers rent a pasture. A grazes $150$ cows for $4$ months, B grazes $100$ cows for $6$ months and C grazes $200$ cows for $6$ months. What percentage of the rent should C pay?

   Show the answer and solution

   Answer: 50

   Rent is shared in the ratio of cows $\times$ months. A: $150\times4=600$. B: $100\times6=600$. C: $200\times6=1200$. The total is $2400$, and C's part is $\frac{1200}{2400}=\frac12$. So C pays $50\%$ of the rent.

6. Question 6Type the answerWritten by IPMATwayProportion and Variation

   $x$ varies directly as $y^2$ and inversely as $z$. When $y=4$ and $z=8$, $x=6$. Find $x$ when $y=6$ and $z=3$.

   Show the answer and solution

   Answer: 36

   Write $x=\frac{ky^2}{z}$. With $y=4$, $z=8$, $x=6$: $6=\frac{16k}{8}=2k$, so $k=3$. Now put $y=6$, $z=3$: $x=\frac{3\times36}{3}=36$. (Here $y^2$ grows by $\frac{36}{16}$ and $z$ shrinks by $\frac38$, so $x=6\times\frac{36}{16}\times\frac83=36$.)

7. Question 7Type the answerWritten by IPMATwayPartnership

   Leo starts a business with a certain sum. Max joins $2$ months later with twice Leo's sum, and Noah joins $6$ months after the start with the same sum as Leo. Leo, the working partner, first takes $15\%$ of the year's profit; the rest is shared in the ratio of capital $\times$ time. If the profit is ₹$38000$, how much (in ₹) does Leo receive in all?

   Show the answer and solution

   Answer: 15900

   Capital-months: Leo $12L$, Max $2L\times10=20L$, Noah $L\times6=6L$, i.e. $6:10:3$. Leo's salary is $15\%$ of $38000=5700$. The remaining $32300$ gives Leo $\frac{6}{19}\times32300=10200$. Leo receives $5700+10200=15900$.

8. Question 8Multiple choiceAsked in IPMAT Indore 2024Ratios

   For some non-zero real values of $a$, $b$ and $c$, it is given that $\left|\frac ca\right|=4$, $\left|\frac ab\right|=\frac13$ and $\frac bc=-\frac34$. If $ac>0$, then $\frac{b+c}{a}$ equals

   1. A.$7$
   2. B.$-7$
   3. C.$-1$
   4. D.$1$
   Show the answer and solution

   Answer: D. $1$

   The moduli give sizes only. Since $ac>0$, $c=4a$. Then $b=-\frac34c=-3a$, which matches $\left|\frac ab\right|=\frac13$. So $\frac{b+c}{a}=\frac{-3a+4a}{a}=1$ for any non-zero $a$.

9. Question 9Multiple choiceAsked in JIPMAT 2023Ratios

   Statement (I): The difference, the sum and the product of two numbers are in the ratio $1:5:12$. The product of the two numbers is $18$.

   Statement (II): The last digit in the decimal representation of $2^{91}$ is $6$.

   1. A.Both Statement (I) and Statement (II) are true.
   2. B.Both Statement (I) and Statement (II) are false.
   3. C.Statement (I) is true but Statement (II) is false.
   4. D.Statement (I) is false but Statement (II) is true.
   Show the answer and solution

   Answer: B. Both Statement (I) and Statement (II) are false.

   Use one multiplier: difference $t$, sum $5t$, product $12t$. Then the numbers are $3t$ and $2t$ with product $6t^2=12t$, so $t=2$, the numbers are $6$ and $4$ and the product is $24$. Powers of 2 end in $2,4,8,6$ repeating; $2^{91}$ has $91\equiv3\pmod4$, so it ends in $8$.

10. Question 10Multiple choiceWritten by IPMATwayPartnership

    A starts a business with Rs $36{,}000$. B joins at the start with Rs $48{,}000$ and adds Rs $12{,}000$ more after $5$ months. C joins after $4$ months with Rs $54{,}000$. At the end of the year the profit is Rs $1{,}27{,}000$. What is B's share?

    1. A.Rs $50{,}800$
    2. B.Rs $47{,}625$
    3. C.Rs $53{,}848$
    4. D.Rs $55{,}000$
    Show the answer and solution

    Answer: D. Rs $55{,}000$

    Profit is shared in proportion to capital multiplied by time. A: $36000\times12=432000$. B: $48000\times5+60000\times7=240000+420000=660000$. C: $54000\times8=432000$. The ratio is $432:660:432=36:55:36$, with total $127$ parts, so B gets $\frac{55}{127}\times127000=55000$. Using B's initial Rs $48{,}000$ for the whole year gives Rs $50{,}800$, ignoring the amounts gives Rs $47{,}625$, and counting the extra money for the wrong $5$ months gives Rs $53{,}848$.