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Lesson Posted on 28/06/2017 Tuition/Class I-V Tuition Tuition/Class VI-VIII Tuition Tuition/Class IX-X Tuition
Padmini R.
I am an experienced, qualified teacher and tutor with over 15 years of experience in teaching Maths and...
We know, in unitary method, we first find the value of one quantity from the value of the given quantity. Then we use this value to find the value of the required quantity. While working out the problems using unitary method we come across certain variations in which the values of two quantities depend on each other in such a way that a change in one, results in a corresponding change in the other; then the two quantities are said to be in variation and the two types of variation which occur are called direct and inverse variations.
Solved examples of mixed problems using unitary method:
1. If 24 painters working for 7 hours a day, for painting a house in 16 days. How many painters are required working for 8 hours a day will finish painting the same house in 12 days?
Solution:
24 painters working for 7 hours paint a house in 16 days.
1 painter working for 7 hours paints a house in 16 × 24 days.
1 painter working for 1 hour paints a house in 16 × 24 × 7 days.
Let the required number of painters be x, then;
x painters working for 1 hour a day paint the house in (16 × 24 × 7)/x days
x painters working for 8 hours a day paint the house in (16 × 24 × 7)/(x × 8) days
But the number of days given = 12
According to the problem;
(16 × 24 × 7)/(x × 8) = 12
2688/8x = 12
8x × 12 = 2688
96x = 2688
x = 2688/96
x = 28
Therefore, 28 painters working for 8 hours a day will finish the same work in 12 days.
2. 11 potters can make 143 pots in 8 days. How many potters will be required to make 169 pots in 4 days?
Solution:
11 potters can make 143 pots in 8 days.
1 potter can make 143 pots in 8 × 11 days.
1 potter can make 1 pot in (8 × 11)/143 days.
Let the number of potters required be x, then;
x potters can make 1 pot in (8 × 11)/( 143 × x) days
x potters can make 169 pots in (8 × 11 × 169)/(143 × x ) days
But the number of days given = 4
According to the problem;
(8 × 11 × 169)/(143 × x ) = 4
14872/143x = 4
572x = 14872
x = 14872/572
x = 26
Therefore, 26 potters are required to make 169 pots in 4 days.
We will learn ‘what direct variation is’ and how to solve different types of problems on some situations of direct variation.
If two quantities are related in such a way that the increase in one quantity results in a corresponding increase in the other and vice versa, then such a variation is called a direct variation.
If the two quantities are in direct variation then we also say that they are proportional to each other
Suppose, if the two quantities ‘x’ and ‘y’ are in direct variation, then the ratio of any two values of x is equal to the ratio of the corresponding values of y.
i.e., x1x2=y1y2">x1/x2=y1y/2
or, x1y1=x2y2">x1/y1=x2/y2
Some situations of direct variation:
3. A motor bike travels 280 km in 40 liters of petrol. How much distance will it cover in 9 liters of petrol?
Solution:
We will learn ‘what inverse variation is’ and how to solve different types of problems on some situations of inverse variation.
If two quantities are related in such a way that increase in one quantity causes corresponding decrease in the other quantity and vice versa, then such a variation is called an inverse variation or indirect variation.
Problems on different situations of inverse variation:
1. If 48 men can do a piece of work in 24 days, in how many days will 36 men complete the same work?
Solution:
This is a situation of indirect variation.
Less men will require more days to complete the work.
48 men can do the work in 24 days
1 man can do the same work in 48 × 24 days
36 men can do the same work in (48 × 24)/36 = 32 days
Therefore, 36 men can do the same work in 32 days.
2. 100 soldiers in a fort had enough food for 20 days. After 2 days, 20 more soldiers join the fort. How long will the remaining food last?
Solution:
More soldiers, therefore, food lasts for less days.
This is a situation of indirect variation.
Since 20 soldiers join the fort after 2 days, therefore, the remaining food is sufficient for 100 soldiers and 18 days.
1. 12 typists working for 4 hours to type a book in 18 days. In how many days 4 typists will work for 8 hours to type same book?
Solution:
This is a situation of indirect variation.
12 typists working for 4 hours type a book in 18 days
1 typist working for 4 hours types a book in 18 × 12 days.
1 typist working for 1 hour types a book in 18 × 12 × 4 days.
4 typists working for 1 hour type a book in (18 × 12 × 4)/4
4 typists working for 8 hours type a book in (18 × 12 × 4)/(4 × 8) days.
Therefore, 4 typists working for 8 hours type a book in 27 days.
2. 16 men can build a wall in 56 hours. How many men will be required to do the same work in 32 hours?
Solution:
This is a situation of inverse variation
More the number of men, the faster will they build the wall.
In 56 hours, the wall is built by 16 men.
In 1 hour, the wall is built by 16 × 56 men.
In 32 hours, the wall is built by (16 × 56)/32 men
Therefore, in 32 hours, the wall is built by 28 men.
3. If 72 workers can do a piece of work in 40 days, in how many days will 64 workers complete the same work?
Solution:
This is a situation of indirect variation.
Less workers will require more days to complete the work.
72 workers can do the work in 40 days
1 worker can do the same work in 72 × 40 days
64 workers can do the same work in (72 × 40)/64
Therefore, 64 workers can do the same work in 45 days.
read lessLesson Posted on 19/06/2017 Tuition/Class VI-VIII Tuition Tuition/Class VI-VIII Tuition/Mathematics
Multiplication of even number with 6
Ajinkya
Lesson Posted on 18/06/2017 Tuition/Class VI-VIII Tuition Tuition/Class IX-X Tuition Tuition/Class VI-VIII Tuition/Mathematics
Profit and loss important formulas
Padmini R.
I am an experienced, qualified teacher and tutor with over 15 years of experience in teaching Maths and...
IMPORTANT FACTS
Cost Price:
The price, at which an article is purchased, is called its cost price, abbreviated as C.P.
Selling Price:
The price, at which an article is sold, is called itsselling prices, abbreviated as S.P.
Profit or Gain:
If S.P. is greater than C.P., the seller is said to have a profit or gain.
Loss:
If S.P. is less than C.P., the seller is said to have incurred a loss.
IMPORTANT FORMULAE
Gain = (S.P.) - (C.P.)
Loss = (C.P.) - (S.P.)
Loss or gain is always reckoned on C.P.
Gain Percentage: (Gain %) =
(Gain/C.P )×100
5.Loss Percentage: (Loss %)=
Lesson Posted on 16/06/2017 Tuition/Class IX-X Tuition Tuition/Class VI-VIII Tuition Tuition/Class IX-X Tuition/Mathematics
Soumi Roy
I am an experienced,qualified tutor with over 5 years of experience in teaching across many boards including...
1) 6r + 7 = 13 + 7r
2) 13 − 4x = 1 − x
3) −7x − 3x + 2 = −8x − 8
4) −8 − x = x − 4x
5) −14 + 6b + 7 − 2b = 1 + 5b
6) n + 2 = −14 − n
7) n − 3n = 14 − 4n 8) 7a − 3 = 3 + 6a
9) 5 + 2x = 2x + 6
10) −10 + x + 4 − 5 = 7x − 5
11) −8n + 4(1 + 5n) = −6n − 14
12) −6n − 20 = −2n + 4(1 − 3n)
13) 4n − 40 = 7(−2n + 2)
14) 7(5a − 4) − 1 = 14 − 8a
15) −31 − 4x = −5 − 5(1 + 5x)
16) 38 + 7k = 8(k + 4)
17) 8x + 4(4x − 3) = 4(6x + 4) − 4
18) 3(1 − 3x) = 2(−4x + 7)
19) 4(−8x + 5) = −32x − 26
20) −3(x − 1) + 8(x − 3) = 6x + 7 − 5x
read lessLesson Posted on 15/06/2017 Tuition/Class IX-X Tuition Tuition/Class VI-VIII Tuition Tuition/Class VI-VIII Tuition/Mathematics
Padmini R.
I am an experienced, qualified teacher and tutor with over 15 years of experience in teaching Maths and...
The base b raised to the power of n/m is equal to:
b^{n/m} = (^{m}√b)^{n} = ^{m}√(b^{n})
Example:
The base 2 raised to the power of 3/2 is equal to 1 divided by the base 2 raised to the power of 3:
2^{3/2} = ^{2}√(2^{3}) = 2.828
Fractions with exponents:
(a / b)^{n} = a^{n} / b^{n}
Example:
(4/3)^{3} = 4^{3 }/ 3^{3} = 64 / 27 = 2.37
The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power of n/m:
b^{-n/m} = 1 / b^{n/m} = 1 / (^{m}√b)^{n}
Example:
The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power of 1/2:
2^{-1/2} = 1/2^{1/2} = 1/√2 = 0.7071
The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power of n:
(a/b)^{-n} = 1 / (a/b)^{n} = 1 / (a^{n}/b^{n}) = b^{n}/a^{n}
Example:
The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of 3:
(2/3)^{-2} = 1 / (2/3)^{2} = 1 / (2^{2}/3^{2}) = 3^{2}/2^{2 }= 9/4 = 2.25
Multiplying fractional exponents with same fractional exponent:
a^{ n/m} ⋅ b^{ n/m} = (a ⋅ b)^{ n/m}
Example:
2^{3/2} ⋅ 3^{3/2} = (2⋅3)^{3/2} = 6^{3/2} =√(6^{3}) = √216 = 14.7
Multiplying fractional exponents with same base:
a^{ n/m} ⋅ a^{ k/j} = a^{ (n/m)+(k/j)}
Example:
2^{3/2} ⋅ 2^{4/3} = 2^{(3/2)+(4/3)} = 7.127
Multiplying fractional exponents with different exponents and fractions:
a^{ n/m} ⋅ b^{ k/j}
Example:
2^{3/2} ⋅ 3^{4/3} = √(2^{3}) ⋅^{ 3}√(3^{4}) =2.828 ⋅ 4.327 = 12.237
Multiplying fractions with exponents with same fraction base:
(a / b)^{ n} ⋅ (a / b)^{ m} = (a / b)^{n+m}
Example:
(4/3)^{3} ⋅ (4/3)^{2} = (4/3)^{3+2} = (4/3)^{5} = 4^{5} / 3^{5} = 4.214
Multiplying fractions with exponents with same exponent:
(a / b)^{ n} ⋅ (c / d)^{ n} = ((a / b)⋅(c / d))^{ n}
Example:
(4/3)^{3} ⋅ (3/5)^{3} = ((4/3)⋅(3/5))^{3}= (4/5)^{3} = 0.8^{3} = 0.8⋅0.8⋅0.8 = 0.512
Multiplying fractions with exponents with different bases and exponents:
(a / b)^{ n} ⋅ (c / d)^{ m}
Example:
(4/3)^{3} ⋅ (1/2)^{2} = 2.37 / 0.25 = 9.481
Dividing fractional exponents with same fractional exponent:
a^{ n/m} / b^{ n/m} = (a / b)^{ n/m}
Example:
3^{3/2} / 2^{3/2} = (3/2)^{3/2} = 1.5^{3/2} =√(1.5^{3}) = √3.375 = 1.837
Dividing fractional exponents with same base:
a^{ n/m} / a^{ k/j} = a^{ (n/m)-(k/j)}
Example:
2^{3/2} / 2^{4/3} = 2^{(3/2)-(4/3)} = 2^{(1/6)} =^{6}√2 = 1.122
Dividing fractional exponents with different exponents and fractions:
a^{ n/m} / b^{ k/j}
Example:
2^{3/2} / 3^{4/3} = √(2^{3}) /^{ 3}√(3^{4}) =2.828 / 4.327 = 0.654
Dividing fractions with exponents with same fraction base:
(a / b)^{n} / (a / b)^{m} = (a / b)^{n-m}
Example:
(4/3)^{3} / (4/3)^{2} = (4/3)^{3-2} = (4/3)^{1} = 4/3 = 1.333
Dividing fractions with exponents with same exponent:
(a / b)^{n} / (c / d)^{n} = ((a / b)/(c / d))^{n} = ((a⋅d / b⋅c))^{n}
Example:
(4/3)^{3} / (3/5)^{3} = ((4/3)/(3/5))^{3}= ((4⋅5)/(3⋅3))^{3} = (20/9)^{3} = 10.97
Dividing fractions with exponents with different bases and exponents:
(a / b)^{ n} / (c / d)^{ m}
Example:
(4/3)^{3} / (1/2)^{2} = 2.37 / 0.25 = 9.481
Adding fractional exponents is done by raising each exponent first and then adding:
a^{n/m} + b^{k/j}
Example:
3^{3/2} + 2^{5/2} = √(3^{3}) + √(2^{5}) = √(27) + √(32) = 5.196 + 5.657 = 10.853
Adding same bases b and exponents n/m:
b^{n/m} + b^{n/m} = 2b^{n/m}
Example:
4^{2/3} + 4^{2/3} = 2⋅4^{2/3} = 2 ⋅^{3}√(4^{2}) = 5.04
Subtracting fractional exponents is done by raising each exponent first and then subtracting:
a^{n/m} - b^{k/j}
Example:
3^{3/2} - 2^{5/2} = √(3^{3}) - √(2^{5}) = √(27) - √(32) = 5.196 - 5.657 = -0.488
Subtracting same bases b and exponents n/m:
3b^{n/m} - b^{n/m} = 2b^{n/m}
Example:
3⋅4^{2/3} - 4^{2/3} = 2⋅4^{2/3} = 2 ⋅^{3}√(4^{2}) = 5.04
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