Ratio and Proportion – Basic Mathematical Concept, Formula, Examples, Problems

Ratio & Proportion – Concept

A Ratio is a comparison of two quantities. A Proportion is an equality of two Ratios.
\displaystyle For\text{ }example,\frac{3}{4}~is\text{ }a\text{ }Ratio,~\frac{6}{8}is\text{ }a\text{ }Ratio.\text{ }The\text{ }Ratio~\frac{3}{4}may\text{ }also\text{ }be\text{ }expressed\text{ }as\text{ }3:4.\text{ }The\text{ }equality\frac{3}{4}=\frac{6}{8}~~called\text{ }Proportion

Now we discuss the key terms related to Ratio & Proportion, and their properties

  • Ratio
  • Comparison of Ratio
  • Proportion
  • Continuous Proportion
  • Properties of Proportion

Ratio & Proportion – Concept : Video in English

Ratio – Meaning & Concept

A Ratio is a comparison of the sizes of two or more quantities, expressed in same unit, by division.

Ratios are used to compare values. They tell us how much of one thing there is, compared to another. For example, Ratio of number of Men and Women of a place.

If x and y are two quantities of the same kind (in same units), then the fraction x/y is called the ratio of x to y.

Ratio is written as x : y. Thus, the ratio of x to y = x/y or x : y.

The quantities x and y are called the terms of the ratio, x is called the first term or antecedent and y is called the second term or consequent.

For example, in the ratio 7 : 8, 7 & 8 are called terms of the ratio. 7 is called first term and 8 is called second term.

The value of a Ratio remains same when both terms of the ratio are multiplied or divided by same (non zero) number. Such Ratio are called Equivalent Ratio.

A Ratio is normally expressed in lowest terms (where the 1st and 2nd term has minimum value).

For example the values of ratio \frac{2}{5}, \frac{4}{{10}}, \frac{8}{{20}} are same (Equivalent). However, such ratio are normally expressed in minimal terms, i.e 2/5 (or 2:5)

The order of terms cannot be changed. For example 2:3 is not same as 3:2

The two numbers in the Ratio must represent same unit. Number of Students and Age of students (different units) cannot be expressed in ratio. However the of number of boys and number of girls in class can be  represented by Ratio

Example : Compute ratio of 1 hour and 30 minutes. Though the ratio are in 2 different units (Hour and Minutes), both of them represent Time, and one can be converted to another to make the unit same, and then compute the ratio.

1 Hour = 60 Minutes. So ratio of  1hour : 30 minutes = 60:30= 2:1

Two common types of ratios are used, like,  part to part and part to whole. For example, in a Dough:

  • The ratio of Flour to Water is a part-to-part ratio. It compares the amount of the two ingredients (Flour and Water).
  • The ratio of  Flour to Dough is a part-to-whole ratio. It compares the amount of one ingredient (Flour) to the sum of all ingredients (Dough).
  • In a school, there are 100 Junior level students and 200 Senior level students. So Junior (100) to senior (200) students ratio (100:200) is part ratio. Ratio of Junior (100) students to total students (100+200=300) is part to whole ratio (100:300).

Ratio : Meaning and Concept – Hindi Video

Types of Ratio

  • Compounded Ratio: The ratio of the product of antecedents to the product of the consequents of two or more ratios is called Compounded Ratio.

Ex: For ratios x : y and u : v the compounded ratio is x x u : y x v or xu : yv Thus compounded ratio of 3 : 4 and 5 : 7 is 3 x 5 : 4 x 7 or 15 : 28.

  • Duplicate Ratio: The ratio of squares of its terms is called Duplicate Ratio.

Ex: For ratio x : y, x2 : y2 is the duplicate ratio. Thus duplicate ratio of 5 : 7 is 52 : 72 or 25: 49.

  • Sub Duplicate Ratio: The ratio of the square roots of the terms of ratio is called as sub duplicate ratio.

Ex: For ratio x : y, √x : √y is the Sub Duplicate Ratio.

  • Triplicate Ratio: The ratio of cubes of the terms of the ratio is called as triplicate ratio.

Ex: For ratio x : y, x3 : y3 is the triplicate ratio. Thus triplicate ratio of 7 : 9 is 73 : 93 or 343 : 729

  • Inverse Ratio or Reciprocal ratio: The ratio of the reciprocals of the terms is called as Inverse Ratio or Reciprocal ratio.

Ex: For ratio x  : y, \frac{1}{x}: \frac{1}{y}, or y:x is  the Inverse or Reciprocal Ratio

Thus inverse or reciprocal ratio of

\displaystyle 5:7\text{ is }\frac{1}{5}:\frac{1}{7}\text{ or 7:5}

Note: The product of a ratio and its inverse is always 1.

Thus 5 : 7 x 7 : 5 = 5 x 7 : 7 x 5 = 35 : 35 = 1.

  • Continued Ratio: The ratio between 3 or more quantities of same kind is called as Continued Ratio.

Ex: For quantities w, x, y, z, w: x: y : z is the Continued Ratio. Thus continued ratio of 3, 5, 7, 8  is 3 : 5 : 7 : 8.

Ex. Find continued Ratio of \frac{3}{5}: \frac{4}{7} : \frac{5}{8}
The LCM of the denominators 5, 7 & 8 is 280. So, multiply both terms of each ratio by same number to make the denominator equal.
\frac{3}{5}=\frac{{3\times 56}}{{5\times 56}} =\frac{{168}}{{280}}, \frac{4}{7}=\frac{{4\times 40}}{{7\times 40}}=\frac{{160}}{{280}}, \frac{5}{8}=\left( {\frac{{5\times 35}}{{8\times 35}}} \right)=\frac{{105}}{{280}} . So, \frac{3}{5}: \frac{4}{7} : \frac{5}{8} = \frac{{168}}{{280}} : \frac{{160}}{{280}} : \frac{{105}}{{280}} = 168:160:105

Types of Ratio : Hindi Video

Comparison of Ratio

Two ratios a:b and c:d may be compared by comparing the multiplied value ad & bc as,

a:b >c:d if ad>bc, a:b <c:d if ad<bc, a:b =c:d if ad=bc

Ex. Compare the ratios (i)7:5 & 5:3, (ii)3:8 & 6:16, (iii)2:3 & 3:5

(i)ad=7×3=21, bc=5×5=25, So, 5:3>7:5 (or 7:5<5:3) , (ii) ad=3×16=48, bc=8×6=48, So, 3:8 = 6:16

(iii) ad=2×5=10, bc=3×3=9, So, 2:3 > 3:5

Comparison of Ratio – Hindi Video

Types of Ratio : Video in English

Ratios – Mathematical Examples- Problems & Solutions

Ex. 1: The Ratio of the no. of Science students to the no. of arts students in a college of 1440 students is 3 : 5. If 36 new arts students are admitted in the college, find how many new Science students may be admitted so that the ratio of the no. of Science students to the no. of arts students may change to 2 : 3.

The Ratio of the no. of Science students to the no. of arts students = 3 : 5.
Sum of the ratios = 3 + 5 = 8
So, the no. of Science students in the college
= \frac{{3~\times ~1440}}{8} = 540
And the no. of arts students  in the college
= \frac{{5\times 1440}}{8} = 900
Let the no. of new Science students admitted be x, then the no. of Science students become (540 + x).
After admitting 36 new arts students , the no, of arts students  become 900 + 36 = 936
According to the problem,
\displaystyle \frac{{540+x}}{{936}}=\frac{2}{3}
Or, 3 (540 + x) = 2 x 936
Or, 1620 + 3x = 1872 or, 3x = 252 or, x = 84.
Hence the no. of new Science students admitted = 84.

Ex. 2: The monthly earning of two women are in the ratio 4 : 5 and their monthly expenditures are in the ratio 7: 9. If each saves Rs.150 per month, find their monthly earnings.

Let the monthly earnings of two women be Rs.4x and Rs.5x so that the ratio is

Rs.4x : Rs.5x = 4 : 5. If each saves Rs.150 per months, then the expenditures of two women are Rs. (4x − 150) and Rs. (5x − 150).
\frac{{4x150}}{{5x-150}} = \frac{7}{9}, or 9(4x-150)=7(5x-150), or 36x − 1350 = 35x – 1050, or, 36x − 35x = 1350 − 1050, or, x = 300
Therefore, the monthly earnings of the two women are Rs.4 x 300 and Rs.5 x 300 i.e. Rs.1200/- and Rs.1500 /-.

Ex. 3: The ratio of the prices of two flat was 16 : 25. Two years later when the price of the first has increased by 20% and that of the second by Rs.545, the ratio of the prices becomes 11 : 20. Find the original prices of the two houses.

Let the original prices of two flat be Rs.16x and Rs.25x respectively. Then by the given conditions,

\displaystyle \frac{{16x+20\%\text{ of 16}x}}{{25x+545}}=\frac{{11}}{{20}}

\frac{{16x~+~3.2x}}{{25x~+~545}} = \frac{{11}}{{20}}

or, 320x + 64x = 275x + 5995

or, 384x − 275x = 5995, or, 109x = 5995; So,  x = 55

Therefore, the original prices of two houses are Rs.16 x 55 and Rs. 25 x 55 i.e. Rs. 880 and

Rs. 1,375.                                                                                         

Ratios Problems & Solution – Hindi Video

Ratios – Problems & Solution- Video in English

Compound Ratio Problems and Solutions Video in Hindi

Proportion – Mathematical Definition and Concept

An equality of two ratios is called a proportion. Four quantities m, n, o, p are said to be in proportion if m: n = o : p (also written as m : n : o: p) i.e. if \frac{m}{n} = \frac{o}{p} i.e. if mp = no.

The quantities m, n, o, p are called terms of the proportion; m, n, o and p are called its first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and Third terms are called Means (or middle terms).

If m : n = o : p then p is called fourth proportional.

Proportions are useful to establish equivalent ratios and solve for unknown quantities.

The unit of elements each ratio in the proportion must be same while the units of the proportion may be different.

10cm:20cm = 2kg:4kg. In this case the units Cm and Kg are different. However, the units of elements each must be same. Here the unit of both the elements in first ratio is cm, and the units of both the elements in second ratio is Kg.

Mathematical Proportion – Cross product rule.

If m : n = o : p are in proportion then \frac{m}{n} = \frac{o}{p} i.e. mp = no

i.e. product of extremes = product of means.

Inverse Proportion

If a ratio is equal to the reciprocal of the other, then either of them is in inverse (or reciprocal) proportion of the other.

\displaystyle For\text{ }example\text{ }\frac{5}{4}~\left( {5:4} \right)~is\text{ }in\text{ }inverse\text{ }proportion\text{ }of\text{ }\frac{4}{5}~\left( {4:5} \right)\text{ }and\text{ }vice-versa.

Direct Proportion & Inverse Proportion – Concept & Examples

  • Direct Proportion : Direct Proportion describes the direct relationship between two quantities, in such a way that, when one increases / decreases, the other also increases / decreases proportionately. Direct proportion is often expressed as x ∝ y. For example, speed of car (x), and distance travelled (y). This means if Speed (x) increases, y (distance travelled) increases proportionately. And, if Speed (x) decrease, y (distance travelled) decrease proportionately.
  • Inverse Proportion : Inverse proportion describes the relationship between two quantities in such a way that, when one increases /decreases , the other decreases/ increases proportionately. Thus, an inverse proportion is often expressed as x ∝ 1/y. For example, speed of car (x), and drive time to travel a specific distance. This means if Speed (x) increases, y (drive time) decreases proportionately, and, if Speed (x) decreases, y (drive time) increases proportionately.

Proportion : Definition and Concept – Hindi Video

Proportion : Definition and Concept – Video in English

Continuous Proportion – Mathematical Definition

Three quantities m, n, o of the same kind (in same units) are said to be in continuous proportion if m : n = n : o i.e. \frac{m}{n} = \frac{n}{o} i.e. n2 = mo

If m, n, o are in continuous proportion, then the middle term n is called the mean proportional between m and o, m is the first proportional and o is the third proportional.

If n is mean proportional, between m and o,
\displaystyle then\text{ }{{n}^{2}}=\text{ }mo\text{ }i.e.\text{ }n=\sqrt{{mo}}.

When three or more numbers are so related that the ratio of the first to the second, the ratio of the second to the third, to the fourth etc., are all equal, the numbers are said to be in continued proportion, and written as
\frac{x}{y} = \frac{y}{z} = \frac{z}{w} = \frac{w}{p} = \frac{p}{q} . When x, y, z, w, p and q are in continued proportion.

Continuous Proportion – Video in English

Continued Proportion Problems and Solutions Video in Hindi

Properties of Proportion – Mathematical Formula

Various Important mathematical properties of Proportion are explained below. These properties are extensively used to solve problems relating to Proportion.

  • Cross Multiplication : If a : b = c : d, then ad = bc
  • Invertendo: If a : b = c : d, then b : a = d : c
  • Alternendo : If a : b = c : d, then a : c = b : d
  • Componendo : If a : b = c : d, then a + b : b = c + d : d
  • Dividendo : If a : b = c : d, then a – b : b = c – d : d
  • Componendo and Dividendo: If a : b = c : d, then a + b : a – b = c + d : c – d
  • Addendo : If a:b=c:d, then a:b=c:d=(a+c):(b+d), or, \frac{a}{b}=\frac{c}{d}=\frac{{a+c}}{{b+d}}
  • Subtrahendo : If a:b=c:d, then a:b=c:d=(a-c):(b-d), or, \frac{a}{b}=\frac{c}{d}=\frac{{a-c}}{{b-d}}

Properties of Proportion : Hindi Video

Properties of Proportion : Video in English

Ratio and Proportion- Quick Calculation Formula

• If u/v = x/y, then uy = vx
• If u/v = x/y, then u/x = v/y
• If u/v = x/y, then v/u = y/x
• If u/v = x/y, then (u+v)/v = (x+y)/y
• If u/v = x/y, then (u-v)/v = (x-y)/y
• If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as componendo -Dividendo Rule
• If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c

Proportions – Problems & Solutions

Ex. 1 : If \frac{x}{6}= \frac{y}{8} = \frac{z}{{14}}, then prove that \frac{{x+y+z}}{z} =2

So, by Addendo, we have, \frac{x}{6}= \frac{y}{8} = \frac{z}{{14}} = \frac{{x~+~y~+~z}}{{6~+~8~+~14}} = \frac{{x~+~y~+~z}}{{28}}

So,\frac{{x~+~y~+~z}}{{28}}= \frac{z}{{14}}. Or by Alternendo , \frac{{x~+~y~+~z}}{z} = \frac{{28}}{{14}} =2

Ex. 2 : What should be added to the terms of 3 : 13 so to get 1 : 3.
Let x be added to each terms of ratio of 3 : 13 to get 1 : 3, or \frac{{3+x}}{{13+x}} = \frac{1}{3}
or by Cross multiplication, 3(3 + x) = (13 + x), or 9 + 3x = 13 + x, or 3x – x = 13 – 9, or 2x = 4, or x = 2.

Ex. 3 : What quantity must be added to each of the terms of the ratio a : b so that it may become equal to c : d ?

Let ‘x’ be added to each term of the ratio a : b to make it c : d.

\displaystyle \frac{{a+x~}}{{b+x}}=\frac{c}{d}
So, by cross multiplication, d(a + x) = c(b + x), or  da + dx = cb + cx, or dx-cx=cb-da,

or  (d – c)x = cb – da,
\displaystyle Or\text{ }x=\frac{{cb-da}}{{d-c}}or\text{ }x=\frac{{ad-bc}}{{c-d}}

Ex. 4 : An ornament weights 15 gm of which 5 gm is pure silver and the rest alloy. Find the ratio of pure silver to alloy?

Weight of ornament = 15 gm. Weight of pure silver = 5 gm. Weight of rest alloy =15 – 5 = 10 gm

\frac{{Weight~of~pure~silver}}{{Weight~of~alloy}} = \frac{5}{{10}} = \frac{1}{2}.
So, Ratio of  Silver to Alloy is  \frac{1}{2}\text{ }\!\!~\!\!\text{ or}~1:2

Ex. 5: If a:b=c:d=5:3, what is the value of ad:bc & (a+c):(b+d)

\frac{a}{b}=\frac{c}{d}. So, ad=bc, so, ad:bc=1:1.(Ans)

Again, \frac{a}{b}=\frac{c}{d}. So, \frac{a}{b}=\frac{c}{d} =\frac{{a+c}}{{b+d}}=\frac{5}{3}. (Ans)

Proportions – Problems & Solutions – Hindi Video

Proportions – Problems & Solutions – Video in English

The entire concept of Ration & proportion and numerical example are summarized

Ratio & Proportion – Complete Video in Hindi

Mathematical Rules of Ratio & Proportion – Videos Playlist – English

Mathematical Rules of Ratio & Proportion – Videos Playlist – Hindi

Mathematical Rules of Ratio & Proportion –e-Book (PDF)

Mathematical Rules of Ratio & Proportion- Multiple Choice Questions