Introduction:
Arithmetic is a branch of mathematics that deals with the properties of counting numbers and fractions and the basic operations to these numbers.
RATIO AND PROPORTION
Ratio: If a and b \[\text{Percentage decrease =}\left( \frac{\text{Decrease in quantity}}{\text{Original quantity}}\text{ }\!\!\times\!\!\text{ 100} \right)\text{ }\!\!%\!\!\text{ }\] are two quantities of the same kind, then the fraction \[\left\{ \frac{x}{(100+x)}\times 100 \right\}%\] is called the ratio of a to b and written as a: b, read as a is to b also a is called the antecedent or first term and b is called consequent or second term.
Proportion: Four (non-zero) quantities of the same kind a, b, c and d are said to be in proportion if the ratio of a fob is equal to the ratio of c to d
i.e., if \[\left\{ \frac{x}{(100-x)}\times 100 \right\}%\]
We can write as a: b:: c : d
a, b, c, d are in proportion if ad = bc
a and d are called extreme terms and b and c are cutted middle terms or mean terms.
- The (non-zero) quantities of the same kind a, b, c, d, e, f,... are said to be in continued proportion.
if \[=\left\{ \left( \frac{r}{r+100} \right)\times 100 \right\}%\]
- If a, b, c are in continued proportion, then b is called mean proportional of a and c.
\[=\left\{ \left( \frac{r}{r-100} \right)\times 100 \right\}%\]\[S.P.-C.P\] \[S.P.\text{ }>\text{ }C.P.\]\[C.P.-S.P\] \[C.P.\text{ }>\text{ }S.P.\]
- If a, b, c are in continued proportion then c is called the third proportional.
- If \[Gain\text{ }%=\frac{Gain\times 100}{C.P.},\] then, each ratio is equal to \[Loss%=\frac{Loss\times 100}{C.P.}\]
- If \[S.P.=\frac{100+gain%}{100}\times C.P.\] then \[dsfdsfsd\] (invertendo)
- If \[priyanka\,\,vishwkarma\] then \[12k+8k+12k=2400\] (Allemande)
- If \[S.P.=\frac{100+gain%}{100}\times C.P.\] then \[S.P.=\frac{100+gain%}{100}\times C.P.\] (Componendo)
- If \[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \frac{\text{100 - Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\] then \[{{a}_{13}}=450-372=78\](Dividendo)
- If \[\frac{1}{4}\] then \[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \frac{\text{100 - Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\](Componendo and Dividendo)
PERCENTAGE
The word 'per cent' is an abbreviation of the Latin phrase 'per centum 'which means per hundred or hundredths.
Thus, the term percent means per hundred or for every hundred.
So, \[{{a}_{13}}=450-372=78\]
By a certain per cent we mean that many hundredths.
Important Formula
- To convert a given percentage to a fraction or decimal, divide it by 100 and remove the sign %.
- To convert a given fraction or decimal into percentage, multiply it by 100 and put the sign %. \[{{v}_{1}}=15\]
- \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\]
- \[220=200{{\left( 1+\frac{R}{100} \right)}^{n}}\]
- If A's income is x % more than that of B. Then B's income is less than that of A by \[1+\frac{R}{100}=\frac{220}{200}\]
- If A's income is x % less than that of B. Then B's income is more than that of A by \[R=10%\]
- If the price of an item is increased by r %, then the reduction in consumption, so that expenditure is not increased, \[CI-SI=\frac{R\times SI}{2\times 100}\]
- If the price of commodity decreases by r%, then the increase in consumptions, so that expenditure remains the same, \[144=\frac{15\times SI}{200}\]
PROIT AND LOSS
Cost Price: The price at which an article is made is known as its cost price.
The cost price is abbreviated as C.P.
Selling Price: The price at which an article is sold is known as its Selling price.
The selling price is abbreviated as S.P.
Profit: If the selling price (S.P.) of an article is greater than the cost price (C.P), then the difference between the selling price and cost price is called Profit.
Loss: If the selling price (S.P.) of an article is less than the cost price (C.P), the difference between the cost price (C.P.) and the selling price (S.P.) is called Loss.
IMPORTANT FORMULA
- Gain =\[SI=Rs.1920\], if \[\frac{PTR}{100}=Rs.1920\]
- Loss =\[\frac{P\times 2\times 15}{100}=1920\], if \[P=Rs.6400.\]
- \[A=8000{{\left( 1+\frac{5}{2\times 100} \right)}^{2}}=Rs.8405\]\[CI=A-P=8405-8000=Rs.405.\]
- \[=25000\times \left( 1+\frac{4}{100} \right)\times \left( 1+\frac{5}{100} \right)\times \left( 1+\frac{8}{100} \right)\]
- \[=25000\times \frac{26}{25}\times \frac{21}{20}\times \frac{27}{25}=29,484.\]
- When the selling price and gain per cent are given,
\[=1,75,000\times {{\left( 1-\frac{20}{100} \right)}^{3}}\]
- When the selling price and loss per cent are given
\[=1,75,000\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}=Rs.89,600\]
- If an article receives a gain of x% and other a loss of. x%, then overall % loss \[P\left[ {{\left( 1+\frac{10}{100} \right)}^{2}}-1 \right]=525\], when both articles sold at same price.
- If an artical receives a gain of x% and other a less of x% then overall% less \[P\times \frac{21}{100}=525\]when both articles sold at same price.
DISCOUNT
Discount means reduction in the price. This reduction is always given on the marked price (M.P.) or list price or advertised price.
IMPORTANT FORMULAE
- When discount is offered on an article, then we calculate the selling price (S.P.) as:
S.P. = Marked price - Discount.
- Discount = M.P. - S.P. = Marked price - Selling price
- \[\Rightarrow \]
- P. = M.P. - Discount = M.P. \[P=Rs.2500\]
\[SI=\frac{2500\times 4\times 5}{100}=Rs.500\]
\[2P=P{{\left( 1+\frac{R}{100} \right)}^{3}}\] \[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \frac{\text{100}-\,\text{Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\]
- \[1+\frac{R}{100}={{2}^{\frac{1}{3}}}\]
- Two successive discounts of x% and y% allowed on an item are equivalent to a single discount of
\[16P=P{{\left( 1+\frac{R1}{100} \right)}^{n}}\]which is less than the sum of individual discounts.
- If an article marked x% more than last price and then x% discount is allowed on marked price, then total loss% Two discounts of 15% and 4% are equivalent to a single discount of
\[1+\frac{R}{100}={{2}^{\frac{4}{n}}}\]
SIMPLE INTEREST
- Whenever we borrow money from some lending source such as a bank or a financial institution, we have to pay some extra money to the service provider, which depends upon the sum borrowed by us and the period of time for which we wise to borrow it. This extra money is called the The borrowed money is called principal.
- On the other hand, when we deposit money in a bank for safe keeping, we earn interest. Interest is calculated according to an agreement which specifies the rate of interest. Generally the rate of interest is taken as "per cent per annum" which means " per Rs 100 per year." For example, a rate of' 10% per annum9, means Rs 10 on Rs 100 for 1 year.
- When interest is calculated simply on the original principal, it is known as Simple interest. SI is calculated uniformly on original principle throughout the time period. When the interest for a specific period is added to the principal, then the sum is called the Amount.
Important Formulae
- \[{{2}^{\frac{1}{3}}}={{2}^{\frac{4}{n}}}\]
- \[\frac{1}{3}=\frac{4}{n}\]
- \[1\text{ }year=614.55-578.40=Rs.36.15.\]
- \[R=\frac{100\times I}{P\times T}=\frac{100\times 3.615}{578.40\times 1}=6\frac{1}{4}%\]
S.I. = Simple interest, P = Principal amount, R = Rate of interest, T= Time
COMPOUND INTEREST
- If the interest earned of a specific period is added to the principal for calculating the interest for the next period and so on, then such calculated interest is called Compound interest (C.I.).
Important Formula
- If A is the amount, P is the principal, R% is the rate of interest compounded annually and n is the number of years. Then
\[85\times \frac{12}{20}=51\]
\[\frac{51-42}{42}\times 100=21\frac{3}{7}%\]
- If the interest is compounded half-yearly, then Rate \[=\frac{10}{100}\times 65=6.50\] per half-year and time = 2n half-years.
So, \[\therefore \]
- If interest compounded quarterly, then Rate \[SP=CP-loss=65-6.50=58.50.\] per quarter, Time = 4n quarters.
So, \[=5\times 24=Rs.120.\]
- Let P be the principal and the rate of interest be R% per annum. If the interest is compounded A-times in a year, then the amount A and the compound interest C.I. at the end of n years are given by respectively.
\[=6\times 25=Rs.150\] and \[a%=\frac{a}{100}\]\[\frac{a}{b}=\left( \frac{a}{b}\times 100 \right)%\]
- Let P be the principal and the rate of interest be \[\text{Percentage increase=}\left( \frac{\text{Increase in quantity}}{\text{Original quantity}}\text{ }\!\!\times\!\!\text{ 100} \right)\text{ }\!\!%\!\!\text{ }\] for first year, \[\text{Percentage decrease =}\left( \frac{\text{Decrease in quantity}}{\text{Original quantity}}\text{ }\!\!\times\!\!\text{ 100} \right)\text{ }\!\!%\!\!\text{ }\] for second year, \[\left\{ \frac{x}{(100+x)}\times 100 \right\}%\] for third year and so on and in last \[\left\{ \frac{x}{(100-x)}\times 100 \right\}%\] for the nth year. Then, the amount A and the compound interest C.I. at the end of n years are given by respectively.
\[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \frac{\text{100 - Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\]and,
\[{{a}_{13}}=450-372=78\]
- Let P be the principal and there rate of interest be R% per annum.
- If the interest is compounded annually but time is the fraction of a year, say \[{{v}_{1}}=15\]years, then amount A is given by \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\]and, C.L=A-P.
- If \[220=200{{\left( 1+\frac{R}{100} \right)}^{n}}\]is the value of an article at accrete in time and R% perineum is the rate of depreciation, then the value \[1+\frac{R}{100}=\frac{220}{200}\] at the end of years is given by
\[R=10%\]
- If \[CI-SI=\frac{R\times SI}{2\times 100}\] is the value of an article at a certain time and the rate of depreciation is \[144=\frac{15\times SI}{200}\] for first \[SI=Rs.1920\] years, \[\frac{PTR}{100}=Rs.1920\] for next \[\frac{P\times 2\times 15}{100}=1920\] years and so on and \[P=Rs.6400.\] for the last n^ years, then the value at the end of \[A=8000{{\left( 1+\frac{5}{2\times 100} \right)}^{2}}=Rs.8405\] years is given by
\[CI=A-P=8405-8000=Rs.405.\]
Important Formulae for Population
- Let P be the population of a city or town at the beginning of a certain year and the population grows at a constant rate of R% per annum, then
Population after n years \[=25000\times \left( 1+\frac{4}{100} \right)\times \left( 1+\frac{5}{100} \right)\times \left( 1+\frac{8}{100} \right)\]
Population n years ago \[=25000\times \frac{26}{25}\times \frac{21}{20}\times \frac{27}{25}=29,484.\]
- Let P be the population of a city or a town at the beginning of a certain year. If the population grows at the rate of \[=1,75,000\times {{\left( 1-\frac{20}{100} \right)}^{3}}\] during first year and \[=1,75,000\times \frac{4}{5}\times \frac{4}{5}\times \frac{4}{5}=Rs.89,600\] during second year, then
Population after 2 years \[P\left[ {{\left( 1+\frac{10}{100} \right)}^{2}}-1 \right]=525\]
- This formula may be extended for more than 2 years.
- Let P be the population of a city or a town at the beginning of a certain year. If the population decreases at the rate of R% per annum, then
Population after n years \[P\times \frac{21}{100}=525\]
Population n years ago \[\Rightarrow \]
TYPES OF VARIATION
If the values of two quantities depend on each other in such a way that a change in one results in corresponding change in other, then these two quantities one said to be in variation.
Direct Variation: If two quantities a and b are associated in such a way that increase in quantity a loads to corresponding increase in b in the same proportion and vice-versa then a and b are called in direct variation.
- If two quantities a and b vary with each other in such a manner that the ratio \[P=Rs.2500\] (K is positive real no.) or a = bK then, we say that a and b vary directly with each other or a and b are in direct variation.
Constant (K) is called the constant of variation.
If two quantities a and b are in direct variation and \[SI=\frac{2500\times 4\times 5}{100}=Rs.500\] and \[2P=P{{\left( 1+\frac{R}{100} \right)}^{3}}\] are the corresponding values that the quantities take at one point, then
\[1+\frac{R}{100}={{2}^{\frac{1}{3}}}\]Constant (= k, say) .....(i)
Similarly, if \[16P=P{{\left( 1+\frac{R1}{100} \right)}^{n}}\] and \[1+\frac{R}{100}={{2}^{\frac{4}{n}}}\] are the corresponding values at another point, then
\[{{2}^{\frac{1}{3}}}={{2}^{\frac{4}{n}}}\]
From (i) and (ii), we get
\[\frac{1}{3}=\frac{4}{n}\]
\[1\text{ }year=614.55-578.40=Rs.36.15.\]\[R=\frac{100\times I}{P\times T}=\frac{100\times 3.615}{578.40\times 1}=6\frac{1}{4}%\] [By cross-multiplication]
\[85\times \frac{12}{20}=51\]\[\frac{51-42}{42}\times 100=21\frac{3}{7}%\]
\[=\frac{10}{100}\times 65=6.50\] \[\therefore \]
or, \[SP=CP-loss=65-6.50=58.50.\]
Thus, we obtain the following rule:
Rule: If two quantities a and b are in direct variation, the ratio of any two values of a is equal to the ratio of the corresponding values of b.
EXAMPLE 1:
In which of the following tables, a and b vary directly. Also, find the constant of variation if a and b are in direct variation.
|
a
|
4 7 21 28
|
|
b
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12 21 63 84
|
Sol. We have, \[=5\times 24=Rs.120.\]
Thus, the ratio of the corresponding values of a and b is constant and is equal to \[=6\times 25=Rs.150\].
Hence, a and b are in direct variation with the constant o variation equal to \[a%=\frac{a}{100}\].
Inverse Variation: If two quantities x and y vary with each other in such a manner that the product xy remains constant and is positive, then we say that x and y vary inversely as each other or
x varies inversely as y and y varies inversely as x.
Thus, if two quantities a and b vary inversely as each other, then the product xy always remains constant. The product xy is called the constant of variation.
If two quantities a and b vary inversely as each other and \[\frac{a}{b}=\left( \frac{a}{b}\times 100 \right)%\] are the values of b corresponding to the values \[\text{Percentage increase=}\left( \frac{\text{Increase in quantity}}{\text{Original quantity}}\text{ }\!\!\times\!\!\text{ 100} \right)\text{ }\!\!%\!\!\text{ }\]of a respectively, then
\[\text{Percentage decrease =}\left( \frac{\text{Decrease in quantity}}{\text{Original quantity}}\text{ }\!\!\times\!\!\text{ 100} \right)\text{ }\!\!%\!\!\text{ }\]Constant (= k, say ) and, \[\left\{ \frac{x}{(100+x)}\times 100 \right\}%\]
\[\left\{ \frac{x}{(100-x)}\times 100 \right\}%\] \[=\left\{ \left( \frac{r}{r+100} \right)\times 100 \right\}%\]
\[=\left\{ \left( \frac{r}{r-100} \right)\times 100 \right\}%\] \[S.P.-C.P\]
\[S.P.\text{ }>\text{ }C.P.\] \[C.P.-S.P\] or \[C.P.\text{ }>\text{ }S.P.\]
Thus, we obtain the following rule:
RULE: If two quantities a and b vary inversely as each other, then the ratio of any two values of a is equal to the inverse ratio of the corresponding values of b.
EXAMPLE 2:
In which of the following tables, a and b vary inversely:
|
a
|
4 7 21 28
|
|
b
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12 21 63 84
|
Sol. We know that if a and b vary inversely, then the product ab remains same for all values of a and b.
Here,
\[Gain\text{ }%=\frac{Gain\times 100}{C.P.},\] and
\[Loss%=\frac{Loss\times 100}{C.P.}\]
Clearly, the products of the values of a and the corresponding values of b are fixed. So, a and b vary inversely.
TIME AND WORK
Problems related to Time and Work can be easily solved by using unitary method or By using variations or both of them in mixed way.
RULES:
- If a person can do a work in n days then he/she' will do \[S.P.=\frac{100+gain%}{100}\times C.P.\] th of the work in one day, i.e.,
- If A can do a work in 5 days, it means he can do \[S.P.=\frac{100-loss%}{100}\times C.P.\] th of the work in a day.
Similarly, if many persons work together their work for one day, then this work done is the same as the sum of the works that they can separately do in a day.
Pipes and Cisterns
- As you know that a cistern or a water tank is always connected with two types of pipes. One which fills it up and the other which empties it out. The pipe which fills up the cistern is called an inlet and the one which empties it is called an outlet.
- Let an inlet fills up a cistern in 8 hours. Then, in one hour it fills up \[C.P.=\frac{100}{100+gain%}\times S.P.\] th part of it. We can also say that the work done by inlet in 1 hour is \[C.P.=\frac{100}{100-loss%}\times S.P.\]. Similarly, if an outlet empties out cistern in 6 hours, then in one hour it empties out \[=\frac{{{x}^{2}}}{100}%,\] th part of the cistern, i.e., the work done by the outlet in one hour is \[\frac{{{x}^{2}}}{100}%,\].
- The work done by the inlet is always positive whereas the work done by the outlet is always negative.
If \[Discount%=\frac{Discount}{M.P.}\times 100\] persons can do \[\text{-}\frac{\text{Discount }\!\!%\!\!\text{ }\!\!\times\!\!\text{ M}\text{.P}\text{.}}{\text{100}}\] work (or part of work) in \[S.P.=M.P.\times \left\{ 1-\frac{Discount%}{100} \right\}\] days and m^ persons can do \[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \text{1-}\frac{\text{Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\]works (or part of work) in \[\text{S}\text{.P}\text{.=M}\text{.P}\text{. }\!\!\times\!\!\text{ }\left\{ \frac{\text{100 - Discount }\!\!%\!\!\text{ }}{\text{100}} \right\}\] days, then we have a very general formula in the relationship of \[M.P.=\frac{100\times S.P.}{100-Discount%}\]
TIME AND DISTANCE
- \[\left( x+y-\frac{xy}{100} \right)%\]
- Distance travelled \[\left( 15+4-\frac{15\times 4}{100} \right)=19-\frac{60}{100}=19-\frac{3}{5}=19-0.6=18.4%\]
- If two bodies are moving in the same direction with the speeds of u and v km/h starting from the same point, then their relative speed is (u - v) km/hr and it is (u + v) km/h when they are moving in opposite directions.
- If a man can row aboat at the rate of x km/h in still water and if y km/h is the speed of current in the river, then \[S.I.=\frac{P\times R\times T}{100}\] km/h is the speed of the boat in downstream and \[P=\frac{100\times S.I.}{R\times T}\]km/h is the speed of the boat in upstream.
- When a train passes a pole or a man standing on a platform, time taken to cross the man or pole is equal to the time taken to cover its own length with the given speed.
- Time taken to cover the bridge or a platform is the time to cover the sum of lengths of platform or bridge and its own length.
- If two trains start at the same time from two points A and B towards each other and after crossing each other they take a and b hours in reaching B and A, respectively, then
Speed of A: Speed of \[R=\frac{100\times S.I.}{P\times T}\]
For example: If the length of a bridge is 100 m and that of train is 500 m. and the speed of train is 20 m/s, then Time taken to cover the bridge \[T=\frac{100\times S.I.}{R\times P}\]
For example: If a man is standing near the line then the train of 500 m length and of 20 m/s speed will take the time to cross the man \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\]s.
- When a train is passing another train completely, it has to cover a distance equal to the sum of the lengths of the two trains.
- Let the faster train has length x km and slower train has length y km. Let u km/h is the speed of faster train and v km/
- h is the speed of slower train then time taken by faster train to cross the slower train if both running in the same direction
\[C.I.=A-P=P\left\{ {{\left( 1+\frac{R}{100} \right)}^{n}}-1 \right\}\]hrs.
- Relative speed if running in same directions
\[=\frac{R}{2}%\]
Relative speed if running in opposite directions \[A=P{{\left( 1+\frac{R}{2\times 100} \right)}^{2n}}\]
- Time taken by train to cross each other if running in opposite directions \[=\frac{R}{4}%\]hrs.
- Convertion
\[A=P{{\left( 1+\frac{R}{4\times 100} \right)}^{4n}}\] \[A=P{{\left( 1+\frac{R}{100k} \right)}^{nk}}\]
