Comparing Quantities

1. RATIO
The ratio of two quantities of the same kind in the same units is the fraction that one quantity is of the other. Thus, the ratio ‘a’ is to ‘b’ is the fraction $\frac{a}{b}$ written as a : b. In the ratio a:b; we call ‘a’ as the first term or antecedent and b, the second term or consequent.
Eg: In the ratio 4 : 5
We have first term or antecedent = 4
Second term or consequent = 5
The ratio between two quantities of the same kind and in same units is obtained on dividing
the first quantity by the second.
Eg:

Ratio between 100kg and 150kg = 100 : 150 $=\frac{100}{150}=\frac{2}{3}=2:3$
• Ratio is a fraction. It has no units.
• The quantities to be compared for a ratio should be of the same kind. We cannot have ratio
between 100kg and 2ml.
• To find a ratio between two quantities of the same kind, both the quantities should be taken in the same unit.
Eg:

Ratio between 25ml and 1.5ltr = Ratio between 25ml and 1500 ml $=\frac{25}{1500}=\frac{1}{60}\text{ or }1:60$

If each term of a ratio be multiplied or divided by the same non-zero number, the ratio remains the same.

Eg: i) $2:3=\frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6}=4:6$

       ii) $3:9=\frac{3}{9}=\frac{3÷3}{9÷3}=\frac{1}{3}=1:3$

Ratio in Simplest form or in lowest Term
A ratio a : b is said to be in simplest form, if H.C.F of ‘a’ and ‘b’ is 1.
Eg: i) The ratio of 2 : 3 in simplest form,
Since H.C.F of 2 and 3 is 1
ii) The ratio 15 : 25 is not in simplest form
Since H.C.F of 15 and 25 is 5

$15:25=\frac{15}{25}=\frac{5\times 3}{5\times 5}=\frac{3}{5}=3:5$

Rule: To convert a ratio a : b in simplest form, divide ‘a’ and ‘b’ by the H.C.F of a and b.

Comparison of Ratios
Since ratios are fractions, they can be compared similar to the way we compare fractions. i.e., by converting them into equivalent like fractions or by the cross product method.
Types of ratios
1) Compound Ratio

2) Duplicate Ratio

3) Triplicate Ratio

4) Sub-duplicate Ratio

5) Sub-triplicate Ratio

6) Reciprocal Ratio

1. Compound Ratio
The compound ratio of two ratios a : b and c : d is defined as ac : bd.

In other words, compound ratio is the ratio of the product of antecedents to the product of consequents of the given ratios. Similarly, compound ratio of a : b, c : d, e : f, g : h…. is

$a\times c\times e\times g\times \dots :b\times d\times f\times h\times \dots .\text{ or }\frac{a\times c\times e\times g\dots }{b\times d\times f\times h\dots .}$

Example:
The compound ratio of ratios 2 : 3, 5 : 6 and 1 : 7 is $\frac{2\times 5\times 1}{3\times 6\times 7}=\frac{10}{126}$

2. Duplicate Ratio
The duplicate ratio of a : b is the ratio of the squares of a and b i.e., $a^2:b^2$
Example:
The duplicate ratio of 2 : 3 is $2^2:3^2=4:9$

3. Sub – Duplicate Ratio
The sub-duplicate ratio of a given ratio is equal to the ratio of square roots of the antecedent and the consequent of the given ratio.
The sub-duplicate ratio of a : b is $\sqrt{a}:\sqrt{b}$

Example:
The sub-duplicate ratio of 25 : 49 is $\sqrt{25}:\sqrt{49}=5:7$

4. Triplicate Ratio
The triplicate ratio of a given ratio is the ratio of the cubes of the antecedent and the consequent of the given ratio.
The triplicate ratio of a : b is $a^3:b^3$
Example:
The triplicate ratio of 2 : 3 is $2^3:3^3=8:27$.

5. Sub – triplicate ratio
The sub-triplicate ratio of a given ratio is the ratio of the cube roots of the antecedent and the consequent of the given ratio.
The sub-triplicate ratio of a : b is $\sqrt[3]{a}:\sqrt[3]{b} \text{ or }{a}^{\frac{1}{3}}:b^{\frac{1}{3}}$

Example:
The sub-triplicate ratio of 27 : 125 is $\sqrt[3]{27}:\sqrt[3]{125}=3:5$

6. Reciprocal ratio
The reciprocal ratio of a : b is $\frac{1}{a}:\frac{1}{b}$ which is same as b : a.

Example:
The reciprocal ratio of 16 : 25 is $\frac{1}{16}:\frac{1}{25}\text{ or }25:16$

Commensurable Ratios
If the ratio of any two quantities can be expressed exactly by the ratio of two integers, the
quantities are said to be commensurable. Otherwise, they are incommensurable.
Example:
i) 3 : 5 is commensurable ratio.

ii)$\sqrt{2}:1\text{ and }\sqrt{3}:\sqrt{5}$ are incommensurable ratios.

2. PROPORTION
A statement of equality of two ratios is called a proportion.
Eg: We know that 16 : 36 = 4 : 9
We write it as 16 : 36 :: 4 : 9, where the symbol :: stands for ‘is as’
We say that 16, 36, 4, 9 are in proportion.
Thus, four quantities a, b, c, d are said to be in proportion if a : b = c : d

$\text{ i.e., if }\frac{a}{b}=\frac{c}{d}\text{ i.e., if }ad=bc$

Facts about Proportion
1. In a proportion:
i) The first and fourth terms are called extremes
ii) The second and third terms are known as the means
iii) Product of means = product of extremes
2. If a : b : : c : d, then ‘d’ is called the fourth proportional to a, b, c
3. If a : b = b : c, then we say that
i) a, b, c are in continued proportion
i) ‘c’ is the third proportional to ‘a’ and ‘b’
iii) ‘b’ is the mean proportion between ‘a’ and ‘c’ now, we have a : b = b : c

$\frac{a}{b}=\frac{b}{c};ac=b^2;b=\sqrt{ac}$

Hence, mean proportion between a and $c=\sqrt{ac}$

Comparison of two Ratios
To compare two ratios writing them in the fraction from and equating the denominator and then compare
Note:
i) A number ‘x’ can be divided into two parts in the ratio m : n ; then two parts are $\frac{mx}{m+n},\frac{nx}{m+n}$
ii) A number ‘x’ can be divided into three parts in the ratio m : n : p; then three parts are $\frac{mx}{m+n+p},\frac{nx}{m+n+p},\frac{px}{m+n+p}$

3. TYPES OF PROPORTION
1. Direct Proportion
If two quantities (x, y) are so related that an increase or decrease in one causes a corresponding
increase or decrease in other, then they are said to be in direct proportion or direct variation.
Eg: Number of pens of same price, their total cost.
price of 4 pens = Rs. 12
price of 8 pens = Rs. 24
If in two variables x and y; $\frac{x}{y}$ is constant, then x and y are said to be in direct variation. We write this as $x\propto y$ and read ‘x is directly proportional to y’ .
Eg: Suppose when x = 3, 5, 8,…….
                                   y = 9, 15, 24,……

$\frac{x}{y}=\frac{3}{9}=\frac{5}{15}=\frac{8}{24}=\frac{1}{3}$

So ‘x’ and ‘y’ are in direct variation.
2. Inverse Proportion
If two quantities are so related that an increase in the former causes a corresponding decrease in the later or vice versa; then they are said to be in an inverse variation or inverse proportion.
Eg: Number of men engaged in a work and the time they take to finish it.
If two quantities x and y change such that $x\propto \frac{1}{y}$ then ‘x’ and ‘y’ are said to be in inverse variation.
Then x × y = k (constant).
3. Compound Proportion
Some times, change in a quantity depends upon the changes in two or more other quantities in some proportion. The quantity may be in direct proportion with each of the other quantities. The ratio which shows the change in the quantity is the ratio obtained by compounding of the ratios in which the other quantities change. The quantity may be in inverse proportion with all the others. Then the ratio which shows the change in the quantity is the ratio obtained by compounding the inverse ratio of other quantities.
4. Fourth Proportional
If a : b = c : d, then ‘d’ is called the fourth proportional to a, b, c.
5. Third Proportional
If a, b, c are in continued proportion, then ‘c’ is called the third proportional.
Eg: The third proportional of 3, 9 is 27
4. VARIOUS PROPERTIES OF RATIO AND PROPORTION
1. Invertendo property
If a : b :: c : d then b : a :: d : c.

$\text{ Proof }a:b::c:d\Rightarrow \frac{a}{b}-\frac{c}{d}\Rightarrow b:a::d:c$

2. Alternendo property
If a : b :: c : d then a : c :: b : d.

$\text{ Proof }a:b::c:d\Rightarrow \frac{a}{b}=\frac{c}{d}\Rightarrow \frac{b}{a}=\frac{d}{c}\Rightarrow \frac{a}{b}\cdot \frac{b}{c}=\frac{c}{d}\cdot \frac{b}{c}\text{ or }\frac{a}{c}=\frac{b}{d}\Rightarrow a:c::b:d$

3. Componendo property
If a : b :: c : d then (a + b) : b :: (c + d) : d

$\text{ Proof }a:b::c:d\Rightarrow \frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a}{b}+1=\frac{c}{d}+1\text{ or }\frac{a+b}{b}=\frac{c+d}{d}\Rightarrow (a+b):b::(c+d):d$

4. Dividendo property
If a : b :: c : d then (a – b) : b :: (c – d) : d.

$\mathrm{Proof} a:b::c:d\Rightarrow \frac{a}{b}-\frac{c}{d}\Rightarrow \frac{a}{b}-1-\frac{c}{d}-1\text{ or }\frac{a-b}{b}=\frac{c-d}{d}\Rightarrow (a-b):b::(c-d):d$

5. Convertendo property
If a : b :: c : d then a : (a – b) :: c : (c – d).

$$\begin{gathered} \mathrm{Proof} a:b::c:d\Rightarrow \frac{a}{b}=\frac{c}{d} .........\left(1\right) \\ \text{ or }\frac{\mathrm{a}}{\mathrm{b}}-1=\frac{\mathrm{c}}{\mathrm{d}}-1\text{ or }\frac{\mathrm{a}-\mathrm{b}}{\mathrm{b}}=\frac{\mathrm{c}-\mathrm{d}}{\mathrm{d}} ........\left(2\right) \end{gathered}$$

Dividing (1) by the corresponding sides of (2),

$\frac{\frac{a}{b}}{\frac{a-b}{b}}=\frac{\frac{c}{d}}{\frac{c-d}{d}}$

$\text{ or }\frac{a}{a-b}=\frac{c}{c-d}\Rightarrow a:(a-b)::c:(c-d)$

6. Componendo – dividendo property
If a : b :: c : d then (a + b) : (a – b) :: (c + d) : (c – d).

$\text{ Proot }a:b::c:d\Rightarrow \frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a}{b}+1=\frac{c}{d}+1\text{ and }\frac{a}{b}-1=\frac{c}{d}-1$

$\therefore \frac{\mathrm{a}+\mathrm{h}}{\mathrm{b}}=\frac{\mathrm{c}+\mathrm{d}}{\mathrm{d}}\text{ and }\frac{\mathrm{a}-\mathrm{h}}{\mathrm{b}}=\frac{\mathrm{c}-\mathrm{d}}{\mathrm{d}}$

Dividing the corresponding sides,

$\frac{\frac{a+b}{b}}{\frac{a-b}{b}}=\frac{\frac{c+d}{d}}{\frac{c-d}{d}}\text{ or }\frac{a+b}{a-b}=\frac{c+d}{c-d}$

$\Rightarrow (a+b):(a-b)::(c+d):(c-d)$

Writing in algebraic expressions, the componendo – dividendo property gives the following.

$\frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a+b}{a-b}=\frac{c+d}{c-d}$

This property is frequently used in simplification.

7. Equivalent ratio property

$\text{ If }a:b::c:d\text{ then }(a\pm c):(b\pm d)::a:b\text{ and }(a\pm c):(b\pm d)::c:d$

$\text{ Proof }a:b::c:d\Rightarrow \frac{a}{b}=\frac{c}{d}=k\left(\text{ say }\right)$

$\therefore \mathrm{a}=\mathrm{bk},\mathrm{c}=\mathrm{dk}\cdot \text{ Now },\frac{\mathrm{a}\pm \mathrm{c}}{\mathrm{b}\pm \mathrm{d}}=\frac{\mathrm{bk}\pm \mathrm{dk}}{\mathrm{b}\pm \mathrm{d}}=\frac{\mathrm{k}(\mathrm{b}\pm \mathrm{d})}{\mathrm{b}\pm \mathrm{d}}=\mathrm{k}=\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}$

$\therefore (a\pm c):(b\pm d)::a:b\text{ and }(a\pm c):(b\pm d)::c:d$

Algebraically, the property gives the following.

$\frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a}{b}=\frac{c}{d}=\frac{a+c}{b+d}=\frac{a-c}{b-d}$

Similarly, we can prove that

$\frac{a}{b}=\frac{c}{d}\Rightarrow \frac{a}{b}=\frac{c}{d}=\frac{pa+qc}{pb+qd}$

$\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{e}}{\mathrm{f}}\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{e}}{\mathrm{f}}=\frac{\mathrm{a}+\mathrm{c}+\mathrm{e}}{\mathrm{b}+\mathrm{d}+\mathrm{f}}=\frac{\mathrm{ap}+\mathrm{cq}+\mathrm{er}}{\mathrm{bp}+\mathrm{dq}+\mathrm{fr}}$

$\mathrm{For}\mathrm{example} :\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{2\mathrm{a}+3\mathrm{c}}{2\mathrm{b}+3\mathrm{d}}=\frac{5\mathrm{a}-4\mathrm{c}}{5\mathrm{b}-4\mathrm{d}}=\frac{\mathrm{ab}+\mathrm{cd}}{{\mathrm{b}}^2+{\mathrm{d}}^2}, \mathrm{etc}$

$\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{e}}{\mathrm{f}}\Rightarrow \frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{e}}{\mathrm{f}}=\frac{\mathrm{a}+2\mathrm{c}+3\mathrm{e}}{\mathrm{b}+2\mathrm{d}+3\mathrm{f}}=\frac{4\mathrm{a}-3\mathrm{c}+9\mathrm{e}}{4\mathrm{b}-3\mathrm{d}+9\mathrm{f}},\mathrm{etc}$

5. PERCENTAGE AND APPLICATIONS
Percent: It means ‘per hundred’ ‘out of hundred’ we abbreviate percent by P.C and denote it by the symbol %; thus 15 percent is written as 15%.S
Hundredth Part: Out of 100 equal parts, each part is known as its hundredth part.

$\text{ Thus }\frac{5}{100}=5\text{ hundredths; }\frac{11}{100}=11\text{ hundredths and so on. }$

Percentage: By a certain percentage, we mean that many hundredths.

$\text{ Thus, }7\mathrm{\%}=\frac{7}{100},13\mathrm{\%}=\frac{13}{100},61\mathrm{\%}=\frac{61}{100}\text{ etc…. }$

Conversion of Fraction into a Percentage: To express a fraction as a percent, multiply it by 100.
Eg: $\frac{3}{4}=\left(\frac{3}{4}\times 100\right)\mathrm{\%}=(3\times 25)\mathrm{\%}=75\mathrm{\%}$

Conversion of Percentage into a Fraction: To convert a percentage into a fraction. We divide it by 100 and remove the % sign.
Eg: $10\mathrm{\%}=\frac{10}{100}=\frac{1}{10}$

Percent as a Ratio: A percentage can be expressed as a ratio with its second term 100 and first term equal to the given percentage.
Eg: $36\mathrm{\%}=\frac{36}{100}=\frac{9}{25}=9:25$

Conversion of Ratio into a Percentage: To express a given ratio as a percent, convert the given ratio into a fraction, and then multiply the fraction by 100.
Eg: $3:5=\left(\frac{3}{5}\times 100\right)\mathrm{\%}=(3\times 20)\mathrm{\%}=60\mathrm{\%};5:4=\left(\frac{5}{4}\times 100\right)\mathrm{\%}=(5\times 25)\mathrm{\%}=125\mathrm{\%}$

Conversion of given Percentage into a Decimal: To express a given percentage as a decimal, divide it by 100 and then convert it into decimal form.
Eg: $45\mathrm{\%}=\frac{45}{100}=0.45$

$15\mathrm{\%}=\frac{0.15}{100}=\frac{15}{10000}=0.0015$

Conversion of given Decimal into a Percentage: To express a given decimal as a percent, multiply it by 100.
Eg: $0.4=(0.4\times 100)\mathrm{\%}=40\mathrm{\%}$

$0.004=(0.004\times 100)\mathrm{\%}=0.4\mathrm{\%}$

Finding certain Percentage of a given Quantity: Convert the percentage into a fraction and multiply the given quantity with the resulting fraction.
Eg: $20\mathrm{\%}\text{ of }100\mathrm{kg}=\frac{20}{100}\text{ of }100\mathrm{kg};=\left(\frac{20}{100}\times 100\right)\mathrm{kg};=20\mathrm{kg}$

Expressing one quantity as a Percent of the other
Step1: Take both the quantities in the same units
Step2: Taking the quantity to be compared to numerator and the quantity with which it is to be compared as denominator, express it as a fraction.
Step3: Express this fraction as percent by multiplying with 100
Percentage Change

$\text{ Decrease }\mathrm{\%}=\left(\frac{\text{ Decrease in Value }}{\text{ Orginal Value }}\times 100\right)\mathrm{\%}$

$\text{ Increase }\mathrm{\%}=\left(\frac{\text{ Increase in Value }}{\text{ Orginal Value }}\times 100\right)\mathrm{\%}$

The change in value is always calculated on the original value.

6. PROFIT & LOSS
Cost Price: The price at which an article is purchased is called its cost price, abbreviated as C.P. The overhead expenses like sales tax, transportation etc….. are always included in the cost price.
Net C.P of an article = Actual C.P + overhead expenses
Selling Price: The price at which an article is sold is called its selling price, abbreviated as S.P.
Profit (or) Gain: If (S.P) > (C.P), then the seller has a gain (or) profit given by
Gain = S.P – C.P
Gain always reckoned on C.P.
Gain on$\times$100 is called a gain percent.

$\text{ Gain }\mathrm{\%}=\left(\frac{\text{ Gain }}{\text{ C.P }}\times 100\right)$

Loss: If (S.P) < (C.P), then the seller incurs a loss given by
Loss = (C.P.) – (S.P.)
Loss is always reckoned on C.P.
Loss on $\times$100 is called loss percent

$\mathrm{Loss}\mathrm{\%}=\left(\frac{\mathrm{Loss}}{\mathrm{C}.\mathrm{P}}\times 100\right)$

Finding S.P
When C.P. and Gain % (or) loss% are given, S.P is calculated as follows.

•  $\text{ S.P. }=\left(\frac{100+\text{ Gain }\mathrm{\%}}{100}\right)\times \text{ C.P. }$

•  $\text{ S.P. }=\left(\frac{100-\mathrm{Loss}\mathrm{\%}}{100}\right)\times \text{ C.P. }$

Finding C.P
When S.P. and Gain % (or) loss% are given, C.P is calculated as follows

•  $\text{ C.P. }-\frac{100}{(100+\text{ Gain }\mathrm{\%})}\times \mathrm{S}.\mathrm{P}.$

•  $\text{ C.P. }=\frac{100}{(100-\text{ Loss }\mathrm{\%})}\times \text{ S.P. }$

7. SIMPLE INTEREST
Some times, in need, we borrow money from a money-lender or a bank. We promise to pay it back after a specified period of time. At the end of this period, we have to pay the money borrowed along with some additional money in lieu of using another man’s money.
Principle: The money borrowed (or) lent out for a certain period is called the principle (or) the sum.
Interest: The additional money paid by the borrower for having used the lenders money is called interest.
Amount: The total money paid back to the lender is called amount.
Therefore Amount = Principle + Interest
Rate: Interest on $\times$100 for 1 year is called rate percent per annum. (Abbreviated as rate% p.a).
Thus if rate = 5% per annum, then it means that the interest on $\times$100 for 1 year is $\times$5.
If interest is calculated throughout the loan period, then the interest is called simple interest,
abbreviated as S.I.

Formula for Calculating Simple Interest
Let principle = Rs ‘P’
Time = ‘T’ years and
Rate = ‘R’% per annum
Then, simple interest (S.I.) is given by the formula $=\text{ S.I. }=\frac{P\times R\times T}{100}$

$\text{ Then, }\mathrm{P}=\frac{100\times \mathrm{S}.\mathrm{I}}{\mathrm{R}\times \mathrm{T}};\mathrm{R}=\frac{100\times \mathrm{S}.\mathrm{I}}{\mathrm{P}\times \mathrm{T}};\mathrm{T}=\frac{100\times \mathrm{S}.\mathrm{I}}{\mathrm{P}\times \mathrm{R}}$

Note:
i) While calculating the time period between two given dates, the day on which the money is borrowed is not counted for interest while the day on which the money is returned, is counted.
ii) For converting the time in days into years, we always divide by 365, weather it is an ordinary
years (or) a leap year.

8. TIME AND WORK
• If ‘A’ can complete a piece of work in ‘n’ days, then A’s 1 day’s work = $\frac{1}{n}$
• If A’s 1 day’s work = $\frac{1}{n}$, then time taken by A to finish the whole work = n days.
• If a certain number of workers undertake to do a certain job together in a certain number of days for a certain amount, then this amount is distributed among them in the ratio of their one day work.

9. TIME AND DISTANCE
Speed: The distance travelled by a moving body in a unit of time is called its speed.

$\text{ Speed }=\frac{\text{ Distance }}{\text{ Time }};\text{ Distance }=\text{ Speed }\times \text{ Time }$

$\text{ Time }=\frac{\text{ Distance }}{\text{ Speed }}$

$\text{ Speedin }\mathrm{km}/\mathrm{hr}=\frac{\text{ Distance in km }}{\text{ Time in hrs }}$

$\text{ Speed in }\mathrm{m}/\sec =\frac{\text{ Distance in metres }}{\text{ Time in seconds }}$

Conversion of Speed into other Units
i) To convert a speed of km/hr into m/sec; multiply by $\frac{5}{18}$
ii) To convert a speed of m/sec into km/hr; multiply by $\frac{18}{5}$

Uniform and Variable Speeds
Uniform Speed: If a moving body covers equal distance in equal intervals of time, its speed is said to be uniform.

Eg: Suppose a train covers a 60km in first hour, 60km in second hour, 60km in third hour and
so on. We will say that the train is running at a uniform speed of 60km/hr. In this case we
assume that the speed is uniform.
Variable Speed: if a moving body covers unequal distance in equal intervals of time, its speed is said to be variable.
Average Speed: $=\frac{\text{ Total Distance Covered }}{\text{ Total Time Taken }}$

Note:
i) In crossing a stationary object (like pole (or) a standing man etc) a train has to cover a distance equal to the length of the train.
ii) In crossing a platform, a train has to cover a distance equal to the sum of the length of the train and the platform.
Average: The term ‘average’ plays a vital role when it comes to comparing the performance of two (or) more groups of individuals with respect to a certain parameter such as number of marks secured by the student of two (or) more classes, number of runs scored by the players of two (or) more terms and so on…
We define average as $\text{ Average }=\frac{\text{ Sum of Observations }}{\text{ Number of Observations }}$

using the above formula, we may thus find, a single average value for a set of given individual values. Average thus provides us a single mid-value of the individual scores, that serves to represent the score of the group taken as a whole. From the above formula
Sum of given Observations = (Average) × (Number of Observations)