# 7th Class Mathematics Comparing Quantities

Comparing Quantities

Category : 7th Class

COMPARING QUANTITIES

FUNDAMENTALS

A. Ratio and Proportion

•                         Ratio is a method of comparing two quantities of the same kind by division.
•                        The symbol used to write a ratio is ':' and is read as 'is to'.
•                        A ratio is generally expressed in its simplest form.
•                        A ratio does not have any unit, it is only a numerical value.
•                        To express two terms in a ratio, they should be in the same units of measurement.
•                         When two ratios are equal, they are said to be in proportion. The symbol for proportion is ': :' and is read as 'as to'.

For e.g., 2 is to 3 as to 6 is to 9 is written as $2:3::6:9$ or,$\frac{2}{3}=\frac{6}{9}$

•                         If two ratios are to be equal or to be in proportion, their product of means should be equal to the product of extremes.

Example: If $a:b::c:d$ then the statement ad = bc, holds good.

•                         If $a:b$ and $b:c$ are in proportion such that ${{b}^{2}}=ac$ then b is called the mean proportional of $a:b$ and$b:c$.
•                         Multiplying or dividing the terms of the ratio by the same number gives equivalent ratios.

B. Percentage

•                           Another way of comparing quantities is percentage. The word percent means per hundred. Thus 12% means 12 parts out of 100 parts.
•                           Fractions can be converted into percentages and vice - versa.

e.g.,      (i)$2=2\times 100%=40%$

(ii) $25%=\frac{25}{100}=\frac{1}{4}$

•                         Decimals can be converted into percentages and vice-versa.

e.g.,      (i) $0.36=0.36\times 100%=36%$

(ii) $43%=43-=0.43$

•                       If a number is increased by a% and then decreased by a% or is decreased by a% then increased by a%, then the original number decrease by$\frac{{{a}^{2}}}{100}%$.

Elementary question -1

Q.        Price of a book was decreased by 10% and then increased by 10%. If the original price of book is Rs. 100, what is its current price.

Ans.     Step One:

$Rs.100\xrightarrow{decreased}10%$ Rs.100 means

$100-100\times \frac{10}{100}=100-10=90$

Second step:

$Rs.90\xrightarrow{Increased\,\,by\,10%}90+90\times \frac{10}{100}=90+9=99$However, if we apply above formula, we directly get, new price

$=100-\frac{{{10}^{2}}}{100}%$ of $100=100-\frac{1}{100}\times 100=99$

•                          A number can be split into two parts such that one part is P% of the other. Then the two parts are $\frac{100}{100+P}\times$ number and $\frac{P}{100+P}\times$ number.
•                           If the circumference of a circle is increased (or) decreased by P%, then the radius of a circle increases (or) decreases by P%.
•                          Elementary question - 2: The circumference of a circle is 44cm, if the circumference is increased by 50%, find percentage increase in radius.

Ans.:    ${{C}_{1}}=44\,\,cm$ then

${{C}_{2}}={{C}_{1}}+\frac{50}{100}\times {{C}_{1}}=44+22=66$

${{r}_{1}}=\frac{44}{2\pi }=7$           ${{r}_{2}}=\frac{66}{2\pi }=10.5$

Percentage increase in radius

$\frac{{{r}_{2}}-{{r}_{1}}}{{{r}_{1}}}\times 100%=\frac{10.5-7}{7}\times 100%=50%$

•                         Profit Gain = Selling price (S.P.) - Cost Price (C.P.)
•                         Loss$=C.P.-S.\text{ }P.$
•                         Gain %,$=\frac{gain}{C.P.}\times 100%,$

$S.P.=C.P.+Gain=C.P.+C.P.\times \frac{gain}{100}%$

$=C.P.\left[ 1+\frac{gain%}{100} \right]$

In case of loss,$S.P.=C.P.=\left[ 1-\frac{loss%}{100} \right]$

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

$=\left( \frac{100}{100-loss%} \right)\times S.P.$

•                          When we deposit money in banks, banks give interest on money. Interest may be simple interest (called S.I.)
•                         $S.I.=\frac{P.t.r}{100}$

S.I. = Simple Interest

P = Principal

t = Time

r = Rate percent per annum

•                           Amount (A) = Principal + Interest

$=P+\frac{Ptr}{100}=P\left[ 1+\frac{rt}{100} \right]$

•                          $r\times t=100\,\,(n-1)$

Where r = rate percent

t = time

n = The number of times the sum gets multiplied (i.e. doubled, tripled.....etc.)

•                         S.I. is calculated uniformly on the original principal throughout the time period.

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