Ratio and proportion practice problems: set I: preRMO, pRMO or IITJEE foundation maths

Solve the following equations:

  1. \frac{2x^{3}-3x^{2}+x+1}{2x^{3}-3x^{2}-x-1} = \frac{3x^{3}-x^{2}+5x-13}{3x^{3}-x^{2}-5x+13}
  2. \frac{3x^{4}+x^{2}-2x-3}{3x^{4}-x^{2}+2x+3} = \frac{5x^{4}+2x^{2}-7x+3}{5x^{4}-2x^{2}+7x-3}
  3. \frac{(m+n)x-(a-b)}{(m-n)x-(a+b)} = \frac{(m+n)x+a+c}{(m-n)x+a-c}
  4. If a, b, c, d are proportionals, prove that a+d=b+c+\frac{(a-b)(a-c)}{a}
  5. If a, b, c, d, e are in continued proportion, prove that (ab+bc+cd+de)^{2}=(a^{2}+b^{2}+c^{2}+d^{2})(b^{2}+c^{2}+d^{2}+e^{2})
  6. If the work done by (x-1) men in (x+1) days is to the work done by (x+2) men in (x-1) days in the ratio of 9:10, find x.
  7. Find four proportionals such that the sum of the extremes is 21, the sum of the means 19, and the sum of the squares of all four numbers is 442.
  8. Two casks A and B were filled with two kinds of sherry, mixed in the cask A in the ratio of 2:7, and in the cask B in the ratio of 1:5. What quantity must be taken from each to form a mixture which shall consist of 10 litres of one kind and 45 litres of the other?
  9. Nine litres are drawn from a vessel full of wine; it is then filled with water, then 9 litres of the mixture are drawn, and the vessel is again filled with water. If the quantity of wine now in the vessel to the quantity of water in it as 16 to 9, how much does the vessel hold?
  10. If four positive quantities, are in continued proportion, show that the difference between the first and last is at least three times as great as the difference between the other two.
  11. In a certain country, the consumption of tea is five times the consumption of coffee. If a percent more tea and b percent more coffee were consumed, the aggregate amount consumed would be 7c percent more; but if b percent more tea and a percent more coffee were consumed, the aggregate amount consumed would be 3c percent more; compare a and b.
  12. Brass is an alloy of copper and zinc; bronze is an alloy containing 80 percent of copper, 4 percent of zinc, and 16 percent of tin. A fused mass of brass and bronze is found to contain 74 percent of copper, 16 percent of zinc and 10 percent of tin; find the ratio of copper to zinc in the composition of brass.
  13. A crew can row a certain course up stream in 84 minutes; they can row the same course down stream in 9 minutes less than they could row it in still water; how long would they take to row down with the stream?

Cheers,

Nalin Pithwa

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