7th Class Mathematics Comparing Quantities

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.