Unit 4 - Practice Quiz

1 What does a ratio fundamentally represent in mathematics?

2 If four quantities , , , and are in proportion (), which of the following equations must be true?

3 What is the simplest form of the ratio ?

4 Find the fourth proportional to $2$, $4$, and $6$.

5 What is the duplicate ratio of ?

6 In a partnership, if two partners invest equal amounts for the same period of time, how is the profit shared?

7 The profit-sharing ratio in a partnership is fundamentally dependent on which two factors?

8 The ratio of the present ages of A and B is . If the sum of their ages is $30$ years, what is A's present age?

9 P invests twice as much as Q in a business for the same time period. What will be the ratio of their profit shares (P:Q)?

10 If a person's present age is years, what was their age '' years ago?

11 What is the rule of alligation primarily used for?

12 In the rule of alligation, what does the term 'mean price' refer to?

13 If you mix sugar priced at Rs. $10$/kg and Rs. $20$/kg to get a mixture worth Rs. $15$/kg, what is the ratio of the cheaper sugar to the dearer sugar?

14 A $50$ liter mixture of milk and water contains them in a ratio of . How many liters of milk are in the mixture?

15 Let be the cost of a cheaper ingredient, be the cost of a dearer ingredient, and be the mean price. According to the rule of alligation, the ratio of cheaper to dearer quantity is:

16 When a certain quantity of a uniformly blended mixture is removed from a vessel, what happens to the ratio of its ingredients in the remaining mixture?

17 In the standard removal and replacement formula , what does '' represent?

18 A vessel contains $100$ liters of pure milk. If $10$ liters are removed and replaced with water, how much pure milk remains?

19 If a certain volume of a mixture is drawn out and replaced by an equal volume of pure water, what remains constant throughout the process?

20 A container holds $40$ liters of pure acid. $4$ liters of acid are drawn out and replaced with water. What is the amount of pure acid left in the container?

21 Two numbers are in the ratio . If $9$ is subtracted from each, the new numbers are in the ratio . Find the smaller number.

22 A bag contains $50$ P, $25$ P, and $10$ P coins in the ratio , amounting to Rs. $206$. Find the number of $50$ P coins.

23 The ratio of incomes of A and B is and the ratio of their expenditures is . If at the end of the year, each saves Rs. $1600$, what is the income of A?

24 A sum of money is divided among A, B, and C in the ratio . If C gets Rs. $1200$ more than B, what is A's share?

25 What is the fourth proportional to $5, 8, 15$?

26 A is two years older than B who is twice as old as C. If the sum of the ages of A, B, and C is $27$, how old is B?

27 The ratio of the present ages of A and B is . After $5$ years, the ratio becomes . Find A's present age.

28 A and B invest in a business in the ratio . If of the total profit goes to charity and A's share is Rs. $855$, what is the total profit?

29 A starts a business with Rs. $3500$. After $5$ months, B joins with A as his partner. After a year, the profit is divided in the ratio . What is B's contribution?

30 The sum of the ages of a father and son is $45$ years. Five years ago, the product of their ages was $34$. What are the present ages of the father and son respectively?

31 In what ratio must a grocer mix two varieties of tea worth Rs. $60$ per kg and Rs. $65$ per kg so that by selling the mixture at Rs. $68.20$ per kg he may gain ?

32 A merchant has $1000$ kg of sugar, part of which he sells at profit and the rest at profit. He gains on the whole. What is the quantity sold at profit?

33 Two vessels A and B contain spirit and water mixed in the ratio and respectively. Find the ratio in which these mixtures be mixed to obtain a new mixture containing spirit and water in the ratio .

34 A mixture of $40$ litres of milk and water contains water. How much water must be added to make the mixture contain water?

35 The cost of Type 1 rice is Rs. $15$ per kg and Type 2 rice is Rs. $20$ per kg. If both are mixed in the ratio of , then the price per kg of the mixed variety of rice is:

36 A container contains $40$ litres of milk. From this container, $4$ litres of milk was taken out and replaced by water. This process was repeated two more times. How much milk is now contained by the container?

37 $8$ litres are drawn from a cask full of wine and it is then filled with water. This operation is performed three more times (total four times). The ratio of the quantity of wine now left in the cask to that of water is . How much wine did the cask originally hold?

38 A vessel is full of a mixture of methanol and ethanol in which there is ethanol. $10$ litres of the mixture are drawn off and replaced with pure methanol. If the ethanol is now , what is the total capacity of the vessel?

39 From a cask of milk containing $30$ litres, $6$ litres are drawn out and the cask is filled up with water. If the same process is repeated one more time, what is the ratio of milk to water in the resulting mixture?

40 A vessel contains $60$ liters of a mixture of milk and water in the ratio . $12$ liters of the mixture are drawn off and replaced by water. Find the final ratio of milk to water.

41 A precious diamond falls and breaks into four pieces whose weights are in the ratio . The value of the diamond is directly proportional to the square of its weight. If the loss incurred due to the breakage is Rs. 70,000, what was the original value of the unbroken diamond?

42 The ratio of the incomes of A, B, and C is and the ratio of their expenditures is . If A saves of his income, what is the ratio of the savings of A, B, and C?

43 If the same number is subtracted from , and , the resulting numbers are in proportion. What is the mean proportional between and ?

44 A sum of money is divided among A, B, C, and D such that the ratio of the shares of A and B is , B and C is , and C and D is . If the difference between the shares of C and A is Rs. 840, what is the share of D?

45 The ratio of land to water on Earth is . In the Northern Hemisphere, this ratio is . What is the ratio of land to water in the Southern Hemisphere?

46 A says to B, "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. What is the present age of A?

47 A, B, and C started a business with initial investments in the ratio . At the end of every quarter, A halves his capital, B doubles his capital, and C leaves his capital unchanged. If the total profit at the end of the year is Rs. 49,000, find the difference between the profit shares of B and A.

48 A father's present age is 3 times the sum of the ages of his two children. After 5 years, his age will be twice the sum of the ages of his two children. Find the present age of the father.

49 A, B, and C enter into a partnership. A invests Rs. 4000 for the full year. B invests Rs. 6000 initially but withdraws Rs. 2000 after 6 months. C invests Rs. 8000 but only for the last 8 months. The total profit is Rs. 17,200. If A receives 10% of the total profit for managing the business, what is B's total share in the remaining profit?

50 In a family consisting of a father, mother, son, and daughter, the age of the father is 4 times that of the son. The age of the daughter is that of the mother. The wife is 6 years younger than her husband, and the sister is 3 years older than her brother. What is the mother's present age?

51 Three equal glasses are filled with mixtures of milk and water. The ratios of milk to water in the glasses are , , and respectively. If the contents of all three glasses are mixed into a single vessel, what is the ratio of milk to water in the final mixture?

52 Two alloys contain gold and silver in the ratio and respectively. How many kilograms of the first alloy must be melted with 21 kg of the second alloy to form a new alloy that contains equal quantities of gold and silver?

53 In what ratio should water be mixed with milk costing Rs. 40 per liter so as to make a profit of 25% by selling the resulting mixture at Rs. 40 per liter?

54 Two vessels A and B contain mixtures of acid and water in the ratio and respectively. If a mixture is formed by taking quantities from A and B in the ratio and pouring them into a third vessel C, what will be the ratio of acid to water in C?

55 A merchant buys two kinds of tea at Rs. 80 per kg and Rs. 120 per kg. He mixes them and sells the mixture at Rs. 112 per kg, making a profit of 25%. What is the ratio in which he mixed the two kinds of tea?

56 From a cask completely filled with 24 liters of pure wine, a certain amount is drawn out and replaced with water. This process is done 4 times in total. If the ratio of wine left in the cask to water is now , how many liters were drawn out each time?

57 A container holds 50 liters of pure milk. First, 5 liters of milk are taken out and replaced by water. Then, 10 liters of the resulting mixture are taken out and replaced by water. Finally, 15 liters of this mixture are taken out and replaced by water. How much milk is left in the container?

58 From a vessel containing 100 liters of pure wine, 10% is drawn out and replaced with water. This process is repeated until the concentration of wine drops below 70% for the first time. How many times must this process be performed in total?

59 A vessel is full of a mixture of spirit and water in the ratio . What fraction of the mixture must be drawn off and replaced with pure water so that the final ratio of spirit to water becomes ?

60 A 40-liter mixture of milk and water contains 10% water. 8 liters of the mixture is removed and replaced with pure milk. Then, 10 liters of the new mixture is removed and replaced with pure milk. What is the final percentage of water in the mixture?