&<–2/”>a >nbsp;
Ratio
Introduction:
Ratio is the relation which one quantity bears to another of the same kind. The ratio of two quantities a and b is the fraction a/b and we write it as a: b.
In the ratio a: b, we call a as the first term or antecedent and b, the second term or consequent.
Note: The multiplication or division of each term of a ratio by the same non- zero number does not affect the ratio.
Compound Ratio: – It is obtained by multiplying together the numerators for new numerator and denominators for new denominator.
Example 1. If the ratios are 4:3, 15:20, 2:6 and 3:5 find the compound ratio?
Example2. If we divide 4185 into two parts such that they are in ratio 7:2, then find the values of both the parts?
Sol 2. Let the actual variable be 7x and 2x.
So, the 1st part = 7 ×465=3255
The 2nd part = 2 ×465=930
Note:
The ratio of first , second and third quantities is given by
ac : bc : bd
If the ratio between first and second quantity is a:b and third and fourth is c:d .
Similarly, the ratio of first, second, third and fourth quantities is given by
ace : bce : bde : bdf
If the ratio between first and second quantity is a: b and third and fourth is c:d.
Proportion
Introduction:-
Four quantities are said to be proportional if the two ratios are equal i.e. the A, B, C and D are proportion. It is denoted by “::” it is written as A : B : C : D where A and D are extremes and B and C are called means .
Product of the extreme = Product of the means
Direct proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also increases (or decreases).
Example 1. If 5 pens cost Rs 10 then 15 pen cost?
Sol 1. It is seen that if number of pens increases then cost also increases. So,
5 pens: 15 pens:: Rs 10 : required cost
Inverse proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also decreases (or increases).
Example 2.If 10 men can do a work in 20 days then in how many days 20 men can do that work?
Sol 2. Here if men increase then days should decrease, so this is a case of inverse proportion, so
10 men: 20 men :: required days : 20 days
Rule of three: It Is the method of finding 4th term of a proportion if all the other three are given, if ratio is a:b :: c:d then ,
ALLIGATION
Introduction:-
The word allegation means linking. It is used to find:
- The proportion in which the ingredients of given price are mixed to produce a new mixture at a given price.
- The mean or Average value of mixture when the price of the two or more ingredients and the proportion in which they are mixed are given.
Mathematical Formula:
For two ingredient:-
Example 1: If the rice at Rs 3.20 per kg and the rice at Rs 3.50 per kg be mixed then what should be their proportion so that the new mixture be worth Rs 3.35 per kg ?
Sol 1: CP of 1 kg of cheaper rice CP of 1 kg of dearer rice
Hence they must be mixed in equal proportion i.e. 1:1
Example 2: Find out the ratio of new mixture so that it will cost Rs 1.40 per kg from the given three kinds of rice costing Rs 1.20, Rs 1.45 and Rs 1.74?
Sol 2: 1st rice cost = 120, 2nd rice cost = 145 and 3rd rice cost = 174 paisa.
From the above rule: we have,
Therefore, three rice must be mixed in 39: 20: 20 ratios to have a new mixture of rice.
Questions
Level-I
|
.. |
1. A and B together have Rs. 1210. If of A’s amount is equal to of B’s amount, how much amount does B have? | |||||||
|
| 2. | Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is: | |||||||
|
| 3. | A sum of Money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share? | |||||||
|
| 4. | Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats? | |||||||
|
| 5. | In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity of water to be further added is: | |||||||
|
||||||||
|
6. |
The ratio of the number of boys and girls in a college is 7 : 8. If the Percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio? |
|||||||
|
| 7. | Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit’s salary? | |||||||
|
| 8. | If 0.75 : x :: 5 : 8, then x is equal to: | |||||||
|
| 9. | The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is 5 : 8, then the second number is: | |||||||
|
| 10 .If Rs. 782 be divided into three parts, proportional to : : , then the first part is: | ||||||||
|
Answers
- Answer:Option B
Explanation:
| 4 | A | = | 2 | B | |
| 15 | 5 |
| A = | 2 | x | 15 | B | |
| 5 | 4 |
| A = | 3 | B |
| 2 |
| A | = | 3 | |
| B | 2 |
A : B = 3 : 2.
| B’s share = Rs. | 1210 x | 2 | = Rs. 484. | ||
| 5 |
2 .Answer: Option C
Explanation:
Let the third number be x.
| Then, first number = 120% of x = | 120x | = | 6x |
| 100 | 5 |
| Second number = 150% of x = | 150x | = | 3x |
| 100 | 2 |
| Ratio of first two numbers = | 6x | : | 3x | = 12x : 15x = 4 : 5. | ||
3 .Answer: Option C
Explanation:
Let the Shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x – 3x = 1000
x = 1000.
B’s share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.
4 .Answer: Option A
Explanation:
Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.
Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).
| 140 | x 5x | , | 150 | x 7x | and | 175 | x 8x | |||||||
| 100 | 100 | 100 | ||||||||||||
| 7x, | 21x | and 14x. |
| 2 | ||
| The required ratio = 7x : | 21x | : 14x |
| 2 |
14x : 21x : 28x
2 : 3 : 4.
5 .Answer: Option D
Explanation:
| Quantity of milk = | 60 x | 2 | litres = 40 litres. | |
| 3 |
Quantity of water in it = (60- 40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
| Then, milk : water = | 40 | . | ||
| 20 + x |
| Now, | 40 | = | 1 | ||
| 20 + x | 2 |
20 + x = 80
x = 60.
Quantity of water to be added = 60 litres.
6 .Answer: Option C
Explanation:
Originally, let the number of boys and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x) and (110% of 8x).
| 120 | x 7x | and | 110 | x 8x | |||||
| 100 | 100 | ||||||||
| 42x | and | 44x | |
| 5 | 5 | ||
| The required ratio = | 42x | : | 44x | = 21 : 22 | ||
7 .Answer: Option D
Explanation:
Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
| Then, | 2x + 4000 | = | 40 |
| 3x + 4000 | 57 | ||
57(2x + 4000) = 40(3x + 4000)
6x = 68,000
3x = 34,000
Sumit’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.
8 .Answer: Option B
Explanation:
| (x x 5) = (0.75 x 8) x = | 6 | = 1.20 | ||
| 5 | ||||
9 .Answer: Option B
Explanation:
Let the three parts be A, B, C. Then,
| A : B = 2 : 3 and B : C = 5 : 8 = | 5 x | 3 | : | 8 x | 3 | = 3 : | 24 | ||||
| 5 | 5 | 5 | |||||||||
| A : B : C = 2 : 3 : | 24 | = 10 : 15 : 24 |
| 5 | ||
| B = | 98 x | 15 | = 30. | ||
| 49 |
10 .Answer: Option D
Explanation:
Given ratio = : : = 6 : 8 : 9.
| 1st part = Rs. | 782 x | 6 | = Rs. 204 | ||
Level-II
| 11. | The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries? | |||||||||||||||||||||||||||||||||||||||||||||||||||
Answer: Option C
Explanation: Let A = 2k, B = 3k and C = 5k.
|
||||||||||||||||||||||||||||||||||||||||||||||||||||
| 12. | If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number? | |||||||||||||||||||||||||||||||||||||||||||||||||||
Answer: Option C
Explanation:
A : B = 5 : 3. |
| 13. | The fourth proportional to 5, 8, 15 is: | |||||||||||
Answer: Option B
Explanation: Let the fourth proportional to 5, 8, 15 be x. Then, 5 : 8 : 15 : x 5x = (8 x 15)
|
|
14. |
Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is: |
|||||||||||||
Answer: Option B
Explanation: Let the numbers be 3x and 5x.
23(3x – 9) = 12(5x – 9) 9x = 99 x = 11. The smaller number = (3 x 11) = 33. |
|
15. |
In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there? |
|||||||||||||||||||||||||||||
Answer: Option C
Explanation: Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.
Hence, the number of 5 p coins = (3 x 50) = 150. |
,
Mixed ratio and proportion are two important concepts in mathematics. A ratio is a comparison of two quantities by division. A proportion is an equation that states that two ratios are equal.
To solve a proportion, you can cross-multiply. To do this, multiply the numerator of the first ratio by the denominator of the second ratio, and then multiply the denominator of the first ratio by the numerator of the second ratio. If the two products are equal, then the proportion is true.
A mixed number is a number that is made up of a whole number and a fraction. For example, 3 1/2 is a mixed number. To add or subtract mixed numbers, you must first convert them to FRACTIONS with the same denominator. Then, you can add or subtract the fractions as usual.
An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 5/2 is an improper fraction. To convert an improper fraction to a mixed number, you can divide the numerator by the denominator and then round the remainder up to the nearest whole number.
Equivalent fractions are fractions that have the same value. For example, 1/2 and 2/4 are equivalent fractions. To find equivalent fractions, you can multiply the numerator and denominator of a fraction by the same number.
A percent is a number that is written out of one hundred. For example, 50% is equal to 50/100, which is also equal to 0.5. To convert a fraction to a percent, you can multiply the fraction by 100 and then add a percent sign.
A decimal is a number that is written with a decimal point. For example, 0.5 is a decimal. To convert a fraction to a decimal, you can divide the numerator by the denominator.
To convert between fractions, decimals, and percents, you can use the following conversions:
- Fraction to decimal: Divide the numerator by the denominator.
- Decimal to fraction: Write the decimal as a fraction with a denominator of 100.
- Percent to fraction: Divide the percent by 100 and then add a fraction line.
- Percent to decimal: Divide the percent by 100.
Mixed ratio and proportion are used in many different areas of mathematics, including algebra, geometry, and statistics. They are also used in many real-world applications, such as cooking, carpentry, and finance.
In cooking, mixed ratio and proportion are used to determine the amounts of ingredients that need to be used to make a recipe. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, you can use mixed ratio and proportion to determine how much of each ingredient you need to use if you want to make a double batch of the recipe.
In carpentry, mixed ratio and proportion are used to determine the lengths of different pieces of wood that need to be cut to make a piece of furniture. For example, if you want to make a table that is 6 feet long, you can use mixed ratio and proportion to determine how long the legs of the table need to be.
In finance, mixed ratio and proportion are used to determine the amount of interest that will be paid on a loan. For example, if you borrow $10,000 at a 10% interest rate for 5 years, you can use mixed ratio and proportion to determine how much interest you will pay on the loan.
Mixed ratio and proportion are a powerful tool that can be used to solve many different types of problems. By understanding how to use them, you can make your life easier and more efficient.
What is a ratio?
A ratio is a comparison of two quantities. It can be written in a few different ways:
- $a:b$
- $a \div b$
- $\frac{a}{b}$
The first way is the most common way to write a ratio. The second way is a way to write a ratio as a fraction. The third way is the most concise way to write a ratio.
What is a proportion?
A proportion is an equation that states that two ratios are equal. It can be written in a few different ways:
- $a:b = c:d$
- $\frac{a}{b} = \frac{c}{d}$
The first way is the most common way to write a proportion. The second way is a way to write a proportion as an equation.
What is a mixed number?
A mixed number is a number that is made up of a whole number and a fraction. For example, 3 1/2 is a mixed number.
What is a decimal?
A decimal is a number that is written with a decimal point. For example, 0.5 is a decimal.
What is a fraction?
A fraction is a part of a whole. It is written as two numbers, one on top of the other, with a line between them. The number on top is called the numerator, and the number on the bottom is called the denominator.
What is a percentage?
A percentage is a number that is written out of 100. It is written with a percent sign (%). For example, 50% is equal to 50/100, which is equal to 0.5.
What is a rate?
A rate is a comparison of two quantities that have different units. For example, the speed of a car is a rate. It is a comparison of the distance traveled to the time it takes to travel that distance.
What is a unit rate?
A unit rate is a rate that has a denominator of 1. For example, the speed of a car in miles per hour is a unit rate.
What is a direct proportion?
A direct proportion is a relationship between two quantities in which the ratio of the two quantities is always the same. For example, the distance traveled by a car is in direct proportion to the time it takes to travel that distance.
What is an inverse proportion?
An inverse proportion is a relationship between two quantities in which the product of the two quantities is always the same. For example, the speed of a car is in inverse proportion to the time it takes to travel that distance.
What is a linear equation?
A linear equation is an equation that can be written in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.
What is a quadratic equation?
A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.
What is a cubic equation?
A cubic equation is an equation that can be written in the form $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are real numbers and $a \neq 0$.
What is a quartic equation?
A quartic equation is an equation that can be written in the form $ax^4 + bx^3 + cx^2 + dx + e = 0$, where $a$, $b$, $c$, $d$, and $e$ are real numbers and $a \neq 0$.
What is a quintic equation?
A quintic equation is an equation that can be written in the form $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$, where $a$, $b$, $c$, $d$, $e$, and $f$ are real numbers and $a \neq 0$.
What is a polynomial equation?
A polynomial equation is an equation that can be written in the form $p(x) = 0$, where $p(x)$ is a polynomial. A polynomial is an expression that is made up of terms that are multiplied together, and each term is
Sure, here are some MCQs on the topics of ratios and proportions:
-
What is the ratio of 3 to 5?
(A) 3:5
(B) 5:3
(C) 1:2
(D) 2:1 -
What is the ratio of 12 to 18?
(A) 2:3
(B) 3:2
(C) 4:5
(D) 5:4 -
What is the ratio of 1 to 2?
(A) 1:2
(B) 2:1
(C) 3:4
(D) 4:3 -
What is the ratio of 2 to 3?
(A) 2:3
(B) 3:2
(C) 4:5
(D) 5:4 -
What is the ratio of 3 to 4?
(A) 3:4
(B) 4:3
(C) 5:6
(D) 6:5 -
What is the ratio of 4 to 5?
(A) 4:5
(B) 5:4
(C) 6:7
(D) 7:6 -
What is the ratio of 5 to 6?
(A) 5:6
(B) 6:5
(C) 7:8
(D) 8:7 -
What is the ratio of 6 to 7?
(A) 6:7
(B) 7:6
(C) 8:9
(D) 9:8 -
What is the ratio of 7 to 8?
(A) 7:8
(B) 8:7
(C) 9:10
(D) 10:9 -
What is the ratio of 8 to 9?
(A) 8:9
(B) 9:8
(C) 10:11
(D) 11:10 -
What is the ratio of 9 to 10?
(A) 9:10
(B) 10:9
(C) 11:12
(D) 12:11 -
What is the ratio of 10 to 11?
(A) 10:11
(B) 11:10
(C) 12:13
(D) 13:12 -
What is the ratio of 11 to 12?
(A) 11:12
(B) 12:11
(C) 13:14
(D) 14:13 -
What is the ratio of 12 to 13?
(A) 12:13
(B) 13:12
(C) 14:15
(D) 15:14 -
What is the ratio of 13 to 14?
(A) 13:14
(B) 14:13
(C) 15:16
(D) 16:15 -
What is the ratio of 14 to 15?
(A) 14:15
(B) 15:14
(C) 16:17
(D) 17:16 -
What is the ratio of 15 to 16?
(A) 15:16
(B) 16:15
(C) 17:18
(D) 18:17 -
What is the ratio of 16 to 17?
(A) 16:17
(B) 17:16
(C) 18:19
(D) 19:18 -
What is the ratio of 17 to 18?
(A) 17:18
(B) 18:17
(C) 19:20
(D) 20:19 -
What is the ratio of