Ratio & Proportion

Concept

Ratio: A ratio is a comparison of two quantities in the same units. Simply we separate the two quantities in the ratio with a colon (:). Suppose we want to write the ratio of 6 and 21. We can write this as 6:21 or as a fraction 6/21. First term of a ratio is called antecedent, while the second term is known as consequent. Here, 6 is antecedent and 21 is consequent. Now, ratio of 6 to $\displaystyle \ 21 = 6 : 21 = \frac{6}{21} = \frac{2}{7}$

Proportion: A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. Simply, the equality of two ratios is called proportion. As a example, $\displaystyle \frac{5}{7} = \frac{4}{9}$ , Or 5:7=4:9, Or 5×9=7×4, Or $\displaystyle\frac{45}{28}$ , Or 45:28

Q. If the ratio of boys to girls in a class is 3:2, and there are 25 students in total, how many girls are there in the class?

A) 10

To find the number of girls, we can use a proportion: $\displaystyle \frac{3}{2} = \frac{x}{25-x}$ where ( x ) is the number of boys. Cross-multiplying and solving for ( x ), we get: $\displaystyle \frac{3}{2} = \frac{x}{25-x}$ $\displaystyle 2x = 3(25-x)$ $\displaystyle 5x = 75$ $\displaystyle x = 15$ Therefore, there are 15 boys and ( 25-15 = 10 ) girls in the class.

B) 12

C) 15

D) 18

Discuss

Q. A recipe for lemonade requires 6 cups of water and 2 cups of lemon juice. What is the ratio of water to lemon juice in the recipe?

A) 2:6

B) 3:1

To find the ratio of water to lemon juice, we can divide both quantities by the same factor to get an equivalent ratio in simplest form. In this case, we can divide both by 2 to get 6:2=3:1 Therefore, the ratio of water to lemon juice is 3:1.

C) 4:3

D) 6:2

Discuss

Q. A map has a scale of 1:50,000, which means that 1 cm on the map represents 50,000 cm in real life. How many kilometers does 5 cm on the map represent?

A) 0.25 km

B) 2.5 km

To find the real-life distance represented by a map distance, we can multiply the map distance by the scale factor. In this case, we have 5times of 50,000:1=250,000:5 However, this is in centimeters, and we need to convert it to kilometers by dividing by 100,000 (since there are 100 cm in a meter and 1000 m in a km). Therefore, we get 5 cm on the map represents 2.5 km in real life.

C) 3.75 km

D) 4.50 km

Discuss

Q. In a tournament the score of A : B is 9:5, B : C is 15:7 and C : A is 7:27. If C has the score of 28, what will be the A's score?

A) 60

B) 75

C) 96

D) 108

Answer: $\displaystyle \frac{A}{B} = \left(\frac{9}{5}\right)\times\left(\frac{3}{3}\right)=\frac{27}{15}$ . Now, ( A : B :: B : C :: C : A = 27:15::15:7::7:27 ). So, ( A:B:C=27:15:7 ). Thus, A’s score = $\displaystyle \left(\frac{28}{7}\right)\times27=108$

Discuss

Q. A car travels at a constant speed of 60 km/h for 2 hours, then at 80 km/h for another hour. What is the average speed of the car for the whole journey?

A) 66.33 km/h

B) 70.00 km/h

C) 73.33 km/h

To find the average speed of the car for the whole journey, we can use the formula Average speed = total distance/total time. To find the total distance, we can add the distances traveled at each speed: $\displaystyle 60\times2+80\times1=120+80=200 )$ To find the total time, we can add the times spent at each speed: ( 2+1=3 ). Therefore, the average speed of the car for the whole journey is $\displaystyle Average \ speed = \frac{200}{3} = 66.67 \ km/h$ .

D) 73.67 km/h

Discuss

Q. If Marie has twice as much money as Courie has, who has three times as much money as Sunny has. What is the amount of money they have together if Courie has $12 alone?

A) $30

B) $24

C) $40

Let, Sunny has $x, So, Courie has $3x and Marie has $(3x X2)=$6x. Now, some of the money they have together=x+3x+6x=10x. According to question, 3x=$12 Or, x=$4. Hence, Sum of the money=10x=10X$4=$40.

D) $60

Discuss

Q. A room has a floor area of 150 sq. feet. If the length of the room is 5 feet more than that of width, what will be the ratio of length and breadth of the room?

A) 7:9

B) 5:3

C) 2:3

D) 3:2

Let, width of the room=x feet. So, length of the room=x+5 feet. According to question, x(x+5)=150 Or, x2+5x-150=0 Or, x2+15x-10x-150=0 Or, x(x+15)-10(x+15)=0 Or, (x+15)(x-10)=0. As the value of x can’t be negative, x=10. So, Length : Breadth=(x+5) : x=10+5 : 10=15 : 10= 3 : 2.

Discuss

Q. The ratio of gold and silver in an ornament weighing 42gm is 4:3. How much more silver will be needed to make the ratio of gold and silver to be 3:4?

A) 14gm

Amount of gold in the mixture = $\displaystyle 42\times\left[\frac{4}{4+3}\right]=24$ gm and amount of silver = $\displaystyle 42\times\left[\frac{3}{4+3}\right]=18$ gm, Let, silver to be added = ( X ) gm. So, Gold/Silver = $\displaystyle \frac{24}{18+x}=\frac{3}{4}$ , $\displaystyle Gold/Silver = \frac{24}{18+x} = \frac{3}{4}$ $\displaystyle x = 14 \ gm$

B) 12gm

C) 18gm

D) 16gm

Discuss

Q. In a box some red, blue & green marbles are kept in the ratio of 2:3:4. Two more red marbles & four green marbles are added to make the ratio 4:5:8. How many blue marbles were in the box?

A) 25

B) 10

C) 15

Let, X is the common factor. So, Number of Red Marble:Blue Marble:Green Marble=2x:3x:4x. Therefore, Red Marble/Blue Marble= $\displaystyle \frac{2x + 2}{3x} = \frac{4}{5}$ Or,10x+10=12x Or,12x-10x=10 Or,2x=10 Or,x=5. So, number of Red Marbles were 3x=3 X 5 = 15.

D) 12

Discuss

Q. Two partners A and B started a business with 60% and 40% shares respectively. After one year they added a new partner C in the business for a new investment of $5 lac contributing 20% of the share in the business. What is the new share of A and B in the business?

A) 40% and 36%

B) 32% and 48%

C) 25% and 18%

D) 48% and 32%

New share of A= 60%-(60 X 20%)%=60%-12%=48%, B= 40%-(40X20%)%=40%-8%=32% & C=20%.

Discuss

Q. Two types of tea costing $2.80 and $3.20 per kg. are mixed in the proportion of 2:3. What will be the new selling price of the mixture per kg. to make a profit of 50%?

A) $4.56

Let, amount of tea is mixed at 2 kg. and 3 kg. respectively. New price of tea=$(2.80 X 2) + $(3.20 X 3)=$5.6+$9.60=$15.20. Now, to make a profit of 50% new selling price will be $15.20+$(15.20 X 50%)=$15.20+$7.60=$22.80. Selling price of one kg. of tea=$22.80/5=$4.56.

B) $5.19

C) $3.89

D) $5.10

Discuss

Q. Mr. Clerk won a city election where the ratio of his votes & those of his opponent, Mr. Joseph was 4:3. Total number of voters of that city was 1160 of which 85% casted votes. Calculate the margin of votes by which Mr. Joseph was defeated if 6 of the casted votes were rejected by polling agent?

A) 181

B) 156

C) 140

n the question, total number of voters = 1160, Votes casted = $\displaystyle 1160 \times 85% = 986$ , Votes counted = ( 986-6=980 ). In City Election Mr. Clerk’s vote = $\displaystyle 980\times\frac{4}{7}=560$ & Mr. Joseph’s vote = $\displaystyle 980\times\frac{3}{7}=420$ . Therefore, Margin of votes by which Mr. Joseph defeated = ( 560-420=140 ). $\displaystyle Votes \ casted = 986$ $\displaystyle \ Mr \ Clerks \ vote = 560$ $\displaystyle \ Mr \ Josephs \ vote = 420$ $\displaystyle Margin \ of \ votes = 140$

D) 130

Discuss

Q. The production cost of an ounce of serum is divided among materials, labor and overheads in the ratio of 3 : 4 : 1. If the materials cost of an ounce exceeds the cost of overhead by $27.50, what will be the production cost of per ounce serum?

A) $110.00

Let, x be the common factor for cost ratios. So, production cost 3x:4x:x = 8x. According to question, 3x - x = 27.50, or, 2x = 27.50, or, x=13.75. Now, production cost= 8x=8 x 13.75= 110.

B) $87.50

C) $105.10

D) $118.30

Discuss

Q. An amount of $265 is divided among A, B and C in such a way that C gets $110 less than what A gets and B gets half of the amount what A gets. What will be the ratio of their shares?

A) 32:18: 12

B) 15:18:30

C) 30:15:18

B gets half of A. Inversely, A gets twice of B. Let, B gets X. So, A= 2x and C= 2x - 110. Now, A+B+C= x + 2x + 2x -110=5x -110=265, or, 5x=375, or, x=75. Ratio of A : B : C = 150:75:40=30:15:18.

D) 42:30:18

Discuss

Q. In a passenger flight the ratio of business class and economy class fares is 3:2 and that of the ratio of the number of passengers travelled to the flight is 2:5. If total fare collected from the flight is $30,000, what was the amount collected from business class passengers?

A) $10,750

B) $11,250

Answer: The ratio of fares collected from passengers of business class and economy class = 3:2::2:5 = 6:10=3:5. So, the business class fare was $\displaystyle 30,000 \times \left(\frac{3}{8}\right) = \$11,250$

C) $9,500

D) $10,250

Discuss

Q. The monthly income of A and B is in the ratio of 4:5 and that of the expenditure is 8:11. If they saved $400 and $300 respectively, what was their monthly income?

A) $2,000 & $2,500

Answer: Let, Income=X any expenditure=Y. According to question, 4x - 8y = 400------ (i) and 5x - 11y = 300‐------- (ii). Solving the equations we get X= 500 and Y= 200. So, monthly income of A and B is $2000 & $2500 respectively.

B) $2,500 & $3,000

C) $2,500 & $2,000

D) $3,800 & $1,500

Discuss

Q. A Certain amount of money is distributed between A and B in such a way that one-fourth of A and one-fifth of B are in the ratio of 3:1. If the amount is $102, what is the B's portion?

A) $30

Answer: $\displaystyle A \times \left(\frac{1}{4}\right)$ : B , $\displaystyle \times \left(\frac{1}{5}\right) = 3 : 1$ or, $\displaystyle \frac{A/4}{B/5}=3/1$ , or, $\displaystyle \frac{5A}{4B}=\frac{3}{1}$ , or, $\displaystyle \frac{A}{B}=\left(\frac{3}{1}\right) \times \left(\frac{4}{5}\right)=\frac{12}{5}$ . Now, B’s portion = $\displaystyle 102 \times \left(\frac{5}{17}\right)= 30$ .

B) $25

C) $46

D) $60

Discuss

Q. A certain job is accomplished by 4 men, 6 women and 8 children together. Their individual wages are paid in the ratio of 7:4:2 respectively. If the children earn $344 as wages, what will be the wages earned by 4 men together?

A) $620

B) $580

C) $602

Answer: Ratio of Wages of single man:woman:children $\displaystyle =7:4:2$ . So, the ratio of wages of 4 men, 6 women and 8 children $\displaystyle =4\times7:6\times4:8\times2=28:24:16=7:6:4$ . So, $\displaystyle \left(\frac{344}{4}\right)\times7=86\times7=602$ .

D) $675

Discuss

Q. Ages of son and father are in the ratio of 11:19. After eight years, ratio of their ages will be 15:23. What was the present age of son?

A) 26

B) 22

Answer: Let, x be the common factor for both of them. So, (11x+8/19x+8)=15/23, or, 253x+184=285x+120, or, 285x-253x=184-120, or, 32x=64, or, x=2. Present age of son=11x=11×2=22 years..

C) 42

D) 28

Discuss

Q. One year ago the ratio between A's and B's salary was 5:4. Ratios of their individual salaries between last year's and present year's salaries are 3:4 and 3:7 respectively. If B's salary for the present year is $6,300, what will be the salary of A for present year?

A) $9,600

B) $7,500

C) $8,000

D) $6,000

Answer: One year ago ratio of salary, A:B $\displaystyle =5:4$ . So, Present salary of A:B $\displaystyle =\left(\frac{5}{3}\right)\times4:\left(\frac{4}{3}\right)\times7=\left(\frac{20}{3}\right):\left(\frac{21}{3}\right)=20:21$ . Now, Salary of $\displaystyle A=\left(\frac{\$6,300}{21}\right)\times20=\$6,000$

Discuss

Q. A recipe for lemonade requires 6 cups of water and 2 cups of lemon juice. What is the ratio of water to lemon juice in the recipe?

A) 2:6

B) 3:1

C) 4:3

D) 6:2

Discuss

Ratio & Proportion

Concept

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