Ratio and Proportion

1.0Comparison

There are two ways of comparison (i) By taking difference (ii) By division

By taking difference : For example, Leena and Kavita got 45 marks and 90 marks respectively in their science examination. So, marks of Leena are 45 less than the marks of Kavita. This kind of comparison is called comparison by difference.

By division : In the above example, another way to make comparison is to say that Leena got $\frac{45}{90}$ or $\frac{1}{2}$ the marks that Kavita got or Kavita got $\frac{90}{45}=2$, that is, twice the marks that Leena got. This kind of comparison is called comparison by division. A comparison by division is called ratio.

• Ratio can be written as a fraction. Fractions may also be written as ratio. E.g. $\frac{p}{q}$ or $p:q$

2.0Ratio

The relation between two quantities (both of the same kind and in the same unit) obtained on dividing one quantity by the other, is called the ratio. The ratio between two numbers or quantities is denoted by the colon ":". Thus, the ratio between two quantities p and $\mathrm{q}=\mathrm{p}:\mathrm{q}$. The numbers p and q are called the terms of the ratio $\mathrm{p}:\mathrm{q}$. The first term p is the antecedent and second term q is the consequent. Here, p and q are two non-zero numbers.

Note: The ratio p : q has no units. It is independent of the units of p and q .

• The ratio $\mathrm{a}:\mathrm{b}$ is defined only if a and b are non-zero numbers.
• The ratio $\mathrm{p}:\mathrm{q}$ is equal to $1:1$ only when $\mathrm{p}=\mathrm{q}$.
• The ratio $\mathrm{p}:\mathrm{q}$ and $\mathrm{q}:\mathrm{p}$ are not equal. They are equal only if $\mathrm{p}=\mathrm{q}$. So, $\frac{\mathrm{p}}{\mathrm{q}}\ne \frac{\mathrm{q}}{\mathrm{p}}$.

Simplest form of ratio

A ratio is in its simplest form if the terms of the ratio have no common factors other than 1 . This is also called "reduced form" or "lowest term" of a ratio.

Equivalent ratios

If we multiply or divide both the numerator and the denominator by the same number, we get an equivalent ratio. E.g., $\frac{40}{100}=\frac{40÷2}{100÷2}=\frac{20}{50}$, $\frac{2}{5}=\frac{2\times 3}{5\times 3}=\frac{6}{15}$

Comparison of ratios

Steps

(i) Write the given ratios as fractions in the simplest form. The order of terms is very important in a ratio. Thus, the ratio 3 : 4 is entirely different from 4 : 3. (ii) Find the LCM of the denominators of the fractions. (iii) Convert them into like fractions with same denominators. (iv) Compare the numerators and arrange the fractions. (v) Then respective ratio are also in the same order.

• If two ratios are equivalent, their simplest forms are the same.
• Equivalent ratio represent the same proportion or scaling factor.
• Compare the fractions $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{d}}$

$\frac{a}{b}\times \frac{c}{d}$

ad $\square$ bc

If $\mathrm{ad}>\mathrm{bc}$, then $\frac{\mathrm{a}}{\mathrm{b}}>\frac{\mathrm{c}}{\mathrm{d}}$ $\mathrm{ad}<\mathrm{bc}$, then $\frac{\mathrm{a}}{\mathrm{b}}<\frac{\mathrm{c}}{\mathrm{d}}$ $ad=bc$, then $\frac{a}{b}=\frac{c}{d}$

Dividing a number in the given ratio

If the number p is to be divided in the ratio $\mathrm{a}:\mathrm{b}$, then the first part $=\left(\frac{a}{a+b}\right)\times p$ and the second part $\left(\frac{b}{a+b}\right)\times p$ For example, let's divide 35 in the ratio 2:3. First part $=\left(\frac{2}{2+3}\right)\times 35=\frac{2}{5}\times 35=14$; second part $=\left(\frac{3}{2+3}\right)\times 35=\frac{3}{5}\times 35=21$

3.0Proportion

If two ratios are equal, we say they are in proportion. If four numbers $a,b,c$ and $d$ are such that the ratio of the first two is equal to the ratio of the last two, i.e., $a:b=c:d$, then we say $a,b,c$ and d are in proportion. The symbol '::' can also be used to denote equality of two ratios, i.e., $\mathrm{a}:\mathrm{b}::\mathrm{c}:\mathrm{d}$, read as 'a is to b as $c$ is to $d^{\prime }$ also means $a,b,c$ and $d$ are in proportion. Here $a$ and $d$ are called extreme terms and $b$ and c are called mean terms. The number d is also known as the fourth proportional to $\mathrm{a},\mathrm{b}$ and c . For example, 3, 4, 18 and 24 are in proportion because $3:4=18:24$. If $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ are in proportion, which implies, $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$, then $a\times d=b\times c$, i.e., Product of the extremes $=$ Product of the means

So, we can write it as, $3:5:12$ : $20$ Continued proportion Three numbers $\mathrm{a},\mathrm{b},\mathrm{c}$ are said to be in continued proportion if $\mathrm{a},\mathrm{b},\mathrm{b}$ and c are in proportion. i.e., $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{b}}{\mathrm{c}}\Rightarrow {\mathrm{b}}^2=\mathrm{ac}$. If $\mathrm{a},\mathrm{b},\mathrm{c}$ are in continued proportion then b is known as the mean proportion of a and $\mathrm{c},\mathrm{c}$ is known as the third proportion.

• If $\mathrm{a},\mathrm{b}$ and c are in continued proportion then, ratio of first two and last two is same. i.e., $a:b=b:c$
• For every proportion, Product of means = product of extremes
  

4.0Unitary method

Look at this problem. If the cost of 3 pens is ₹ 12 , what will be the cost of 8 pens? The cost of 3 pens is ₹ 12 . So, we know that the cost of one pen will be lesser than ₹ 12 . We can find the cost of one pen by dividing ₹ 12 by 3 . The cost of one pen is ₹ $12÷3=₹4$. If we have to find the cost of 8 pens, we know that it will be more than the cost of one pen, and further that it will be 8 times the cost of one pen. So, we multiply ₹ 4 by 8 and we get ₹ 32 . So, the cost of 8 pens is $₹4\times 8=₹32$. In the process of our calculation, we first find out the value of one unit (in this case the cost of one pen). So, this method of problem solving is called the unitary method.

5.0Numerical Ability

Find the ratio of the following : (i) $36$ minutes to $2$ hours (ii) 42 cm to 5 metres (iii) 16 gm to 2 kg (iv) 34 days to 2 years

• Explanation (i) Change both 36 minutes and 2 hours in same unit. Now, 36 minutes $=36$ minutes 2 hours $=2\times 60$ minutes $=120$ minutes Ratio of 36 min. to 2 hours $=\frac{36\text{ minutes }}{2\text{ hours }}=\frac{36\min }{120\min }=\frac{36}{120}=\frac{36÷12}{120÷12}=\frac{3}{10}$ or $3:10$ (ii) 5 metres $=5\times 100\mathrm{cm}=500\mathrm{cm}$ $\therefore$ Ratio of 42 cm to 5 metres $=\frac{42}{5\times 100}=\frac{42÷2}{500÷2}=\frac{21}{250}$ or $21:250$ (iii) 16 gm to 2 kg $2\mathrm{kg}=2\times 1000\mathrm{gm}=2000\mathrm{gm}$ The ratio of 16 gm to $2\mathrm{kg}=\frac{16}{2000}=\frac{1}{125}$ or $1:125$ (iv) 34 days to 2 years 2 years $=2\times 365$ days $=730$ days The ratio of 34 days to 730 days $=\frac{34÷2}{730÷2}=\frac{17}{365}$ or $17:365$

Express the following ratios in its simplest form : (i) $400:256$ (ii) 194:2000

• Explanation (i) $400:256$ The H.C.F of 400 and 256 is 16 $\therefore 400:256=\frac{400}{256}=\frac{400÷16}{256÷16}=\frac{25}{16}=25:16$ (ii) 194:2000 The H.C.F. of 194 and 2000 is 2 $\therefore 194:2000=\frac{194÷2}{2000÷2}=\frac{97}{1000}=97:1000$

There are $20$ girls and 25 boys in a class. (i) What is the ratio of number of girls to the number of boys? (ii) What is the ratio of number of girls to the total number of students in the class?

• Solution Number of girls in a class $=20$ Number of boys in a class $=25$ (i) Ratio of number of girls to the number of boys $=20:25=4:5$ (ii) Ratio of number of girls to the total number of students in the class $=20:(20+25)$ $=20:45=4:9$

Out of $2,000$ students in a school, $750$ opted for basketball, $800$ opted for cricket and the remaining opted for table tennis. If a student can opt for only one game, find the ratio of: (i) Number of students who opted for basketball to those who opted for table tennis. (ii) Number of students who opted for cricket to those who opted for basketball. (iii) Number of students who opted for basketball to the total number of students.

• Solution Total number of students $=2,000$ Number of students who opted for basketball $=750$ Number of students who opted for cricket $=800$ Number of students who opted for table tennis $=2,000-(800+750)=450$ (i) Number of students who opted for basketball to the number of students who opted for table tennis $=750:450=5:3$ (ii) Number of students who opted for cricket to the number of students who opted for basketball =800:750=16:15 (iii) Number of students who opted for basketball to the total number of students $=750:2000=3:8$

Compare 5: 12 and 3 : 8 .

• Explanation $5:12=\frac{5}{12},3:8=\frac{3}{8}$ LCM of $8,12=24$ $5:12=\frac{5}{12}\times \frac{2}{2}=\frac{10}{24}$ $3:8=\frac{3}{8}\times \frac{3}{3}=\frac{9}{24}$ $10>9\Rightarrow \frac{10}{24}>\frac{9}{24}\Rightarrow 5:12>3:8$

The ratio of the number of girls to the number of boys in a school is $5:8$. In another school the ratio of the number of girls to the number of boys is $7:10$. Which school has a higher ratio of girls?

• Explanation The ratios of girls to boys in the two schools are $5:8$ and $7:10$. Since the number of girls form the numerator in both the cases, the school which has a greater ratio has a higher number of girls. We have two fractions $\frac{5}{8}$ and $\frac{7}{10}$. We can compare these fractions by converting both the fractions into fractions with same denominator. The LCM of 8 and 10 is 40 . $\frac{5}{8}=\frac{5\times 5}{8\times 5}=\frac{25}{40},\frac{7}{10}\times \frac{4}{4}=\frac{28}{40}$ $\frac{28}{40}>\frac{25}{40}$ So, the second school with the ratio 7 : 10 has a higher ratio of girls.

The number of stamps in the collections of Jaya, Soumya, and Mamta are in the ratio $3:4:5$. If Soumya has a collection of 108 stamps, find the number of stamps that Jaya and Mamta each has.

• Solution The number of stamps in the collection of Jaya, Soumya, and Mamta are in the ratio $=3:4:5$ Let the number of stamps with Jaya be 3 x . Then Soumya will have 4 x stamps and Mamta will have 5 x stamps. Given that Soumya's $4\mathrm{x}=108$ stamps $\mathrm{x}=\frac{108}{4}=27$ stamps Number of stamps with Jaya's $=3\mathrm{x}=3\times 27=81$ stamps Number of stamps with Mamta's $=5\mathrm{x}=5\times 27=135$ stamps $\therefore$ Jaya has 81 stamps and Mamta has 135 stamps.

Determine if the following are in proportion or not? (i) $4,24,5,30$ (ii) $10,20,30,40$

• Explanation (i) $4,24,5,30$ Ratio of 4 to $24=\frac{4}{24}=1:6$ Ratio of 5 to $30=\frac{5}{30}=1:6$ Since, 4 : $24=5:30$ Therefore, 4, 24, 5, 30 are in proportions. (ii) $10,20,30,40$ Ratio of 10 to $20=\frac{10}{20}=1:2$ Ratio of 30 to $40=\frac{30}{40}=3:4$ Since, $10:20\ne 30:40$ Therefore, 10, 20, 30, 40 are not in proportion.

Find the value of $x$, If $15:60:\mathbf{x}:20$.

• Solution 15:60::x:20 (Given) $\Rightarrow \frac{15}{60}=\frac{x}{20}$ $\Rightarrow \frac{1}{4}=\frac{x}{20}$ $\Rightarrow \mathrm{x}=5$

If 96 tiles required in filling $3$ rooms, then, how many tiles will be required in filling 9 rooms?

• Explanation $\because$ Number of tiles required in 3 rooms $=96$ tiles $\therefore$ Number of tiles required in 1 room $=\frac{96}{3}=32$ tiles $\therefore$ Number of tiles required in 9 rooms $=32\times 9=288$ tiles

6.0Memory Map