A ratio is the comparison or simplified form of two quantities of the same kind. This relation indicates how many times one quantity is equal to the other; or in other words, ratio is a number, which expresses one quantity as a fraction of the other. E.g. Ratio of 3 to 4 is 3 : 4. In this article we will learn the approach applicable to solve various problems on ratios and proportions. In the next few lines you will go through some important concepts related to ratio problems and the methods you should apply to solve those problems. We would like to mention here that you must solve some Ratio and Proportion worksheets to get expertise in this area, only then you would be comfortable in attempting the questions of this area in the exam.

The numbers forming the ratio are called terms. The numerator, “3”, in this case, is known as the antecedent and the denominator, “4”, in this case, is known as the consequent.

**Equivalent Ratios**Let us divide a Pizza into 8 equal parts and share it between Ram and Sam in the ratio 2:6. The ratio 2:6 can be written as 2/6;2/6 = 1/3 We know that 2/6 and 1/3 are called equivalent fractions. Similarly we call the ratios 2:6 and 1:3 as equivalent ratios.

From a given ratio x : y, we can get equivalent ratios by multiplying the terms ‘x’ and ‘ y ‘by the same non-zero number.*For example*

1 : 3 = 2 : 6 = 3 : 9

4 : 5 = 12 : 15 = 16 : 20**Ratio and Proportion Problems and Solutions**

∴ The equivalent ratios of 4 : 3 are 8 : 6, 12 : 9, 16 : 12, 20 : 15

The total number of parts = 1 + 3 = 4. As 4 parts = Rs. 320

∴ 1 part = 320/4 = 80 ∴ 3 parts = 3 × 80 = Rs. 240

**If a : b is a ratio then:**

✔ Duplicate ratio of (a : b) is (a2 : b2).

✔ Sub-duplicate ratio of (a : b) is (a1/2 : b1/2).

✔ Triplicate ratio of (a : b) is (a3 : b3).

✔ Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).

**PROPORTION**

Proportion is represented by the symbol ‘= ‘or ‘:: ‘

If the ratio a : b is equal to the ratio c : d, then a, b, c, d are said to be in proportion.

Using symbols we write as a : b = c : d or a : b :: c : d- When 4 terms in proportion, then the product of the two extremes (i.e. the first and the fourth value) should be equal to the product of two middle values (i.e. the second and the third value)

As Product of Means = Product of Extremes ∴ 16 : 12, 4 : 3 are in proportion.

Therefore 3 × a = 4 × 12; By dividing both sides by 3, we get the missing term = (4 × 12)/3 = 16

The product of Means is 4 × 16 = 64. Hence the product of Extremes must also be 64

64 can be written as 4 × 16 or 2× 32 etc. Two proportions are 2: 4:: 16 : 32 and16 : 4 :: 16 : 4.

**FOURTH PROPORTIONAL:**

If a : b = c : d, then d is called the fourth proportional to a, b, c.

**THIRD PROPORTIONAL:**a : b = c : d, then c is called the third proportion to a and b.

Must Read Ratio and Proportion Articles

- Ratio and Proportion: Theory & Concepts
- Ratio and Proportion Practice Problems: Level 01

**MEAN PROPORTIONAL:**Mean proportional between a and b is √ab .

So x = √10x1000 = √10000 = 100.

- CONTINUED PROPORTION a, b, c are in
**Continued Proportion**if a : b = b : c. Here b is called the**Mean Proportional**and is equal to the square root of the product of a and c.b^{2}= a × c → b = √ac - a/b = b/c = c/d etc., then a, b, c, d are in Geometric Progression.

Let a/b = b/c = c/d = k, then, c = dk; b = ck and a = bk

Since c = dk, b = dk × k = dk^{2}and a = bk = dk^{2}× k = dk^{3}, implying they are in Geometric Progression.

If the three ratios, a : b, b : c, c : d are known, we can find a : d by the multiplying these three ratios

a/d = a/b × b/c × c/d - If a/b = c/d= e/f , then each of these ratios is equal to (a+c+e)/(b+d+f)
- If a/b = c/d, then b/a = d/c(Invertendo)
- If a/b = c/d, then a/c = b/d (Alternendo)
- If a/b = c/d , then (a+b)/b = (c+d)/d(Componendo)
- If a/b = c/d, then (a-b)/b = (c-d)/d (Dividendo)
- If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d), (Componendo & dividendo)

**DIRECT VARIATION**Two quantities “x” and “y” are said to be in direct variation if an increase in one quantity results in increase in the other quantity and decrease in one results in decrease in the other quantity. If two quantities vary always in the same ratio, then they are in direct variation.

- Distance and Time are in Direct Variation, because more the distance travelled, the time taken will be more (if speed remains the same).
- Principal and Interest are in Direct Variation, because if the Principal is more, the Interest earned will also be more.
- Purchase of Articles and the amount spent are in Direct Variation, because purchase of more articles will cost more money. If two quantities “x” and “y” vary directly in such a way that x/y remains constant and is positive, and this constant is called the constant of variation. If x α y that means x = py where p is proportionality constant x/y = p, then ratio of any two values of “x” is equal to the ratio of corresponding values of “y” Then x
_{1}/x_{2}= x_{2}/y_{2}.

**INVERSE VARIATION:**

If two quantities “x” and “y” are such that an increase or decrease in “x” leads to a corresponding decrease or increase in “y” in the same ratio, then we can say they vary indirectly or the variation is inverse. Suppose 6 men can do a piece of work in 18 days, then 12 men can do the same job in 9 days. That means if we double the number of men, then number of days get halved. That means there is inverse relation between number of men and number of days.

In general, when two variables x and y are such that xy = k where k is a non-zero constant, we say that y**varies inversely**with x. In notation, inverse variation is written as y α 1/x → y = p/x, where p is constant of proportionality → xy = p. So x_{1}y_{1}= x_{2}y_{2}.

**Examples for Inverse Variation:**

- Work and Time are in Inverse Variation, because more the number of the workers, lesser will be the time required to complete a job.
- Speed and Time are in Inverse Variation, because higher the speed, the lower is the time taken to cover a distance.
- Population and Quantity of food are in Inverse Variation, because if the population increases, the food availability decreases.

a) Form an equation connecting x and y.

b) Calculate the value of y when x = 18.