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Sia ? 6 years, 6 months ago
Let the numerator and the denominator of the fraction be x and y respectively.
Then the fraction is {tex}\frac{x}{y}{/tex}.
Given, The sum of the numerator and the denominator of the fraction is 3 less than the twice of the denominator.
Thus, we have
{tex}x + y = 2y - 3{/tex}
{tex}\Rightarrow{/tex} {tex}x + y - 2y + 3 = 0{/tex}
{tex}\Rightarrow{/tex} {tex}x - y + 3 = 0{/tex}..............(1)
Also given, If the numerator and the denominator both are decreased by 1, the numerator becomes half the denominator. Thus, we have
{tex}x - 1 = \frac{1}{2}(y - 1){/tex}
{tex}{/tex}
{tex}\Rightarrow{/tex} {tex}2(x - 1) = (y - 1){/tex}
{tex}\Rightarrow{/tex} 2x - 2 = (y - 1)
{tex}\Rightarrow{/tex} 2x - y - 1 = 0.......................(2)
So, we have formed two linear equations in x & y as following:-
x - y + 3 = 0
2x - y - 1 = 0
Here x and y are unknowns.
We have to solve the above equations for x and y.
By using cross-multiplication method , we have
{tex}\frac{x}{{( - 1) \times ( - 1) - ( - 1) \times 3}}{/tex} {tex}= \frac{{ - y}}{{1 \times ( - 1) - 2 \times 3}}{/tex} {tex}= \frac{1}{{1 \times ( - 1) - 2 \times ( - 1)}}{/tex}
{tex}\Rightarrow \frac{x}{{1 + 3}}{/tex} {tex}= \frac{{ - y}}{{ - 1 - 6}} = \frac{1}{{ - 1 + 2}}{/tex}
{tex}\Rightarrow \frac{x}{4} = \frac{{ - y}}{{ - 7}} = \frac{1}{1}{/tex}
{tex}\Rightarrow \frac{x}{4} = \frac{y}{7} = 1{/tex}
Using Part I & III , we get x = 4
& From part II & III, we get y = 7 {tex}{/tex}
{tex}\Rightarrow{/tex} x = 4, y = 7
Hence, The fraction is {tex}\frac{x}{y}{/tex} = {tex}\frac{4}{7}{/tex}
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Sia ? 6 years, 6 months ago
1200 = {tex}4 \times 3 \times ( 2 \times 5 ) ^2{/tex}
{tex}= 3 \times 2 ^ { 2 } \times 2 ^ { 2 } \times 5 ^ { 2 }{/tex}
{tex}= 3 ^ { 1 } \times \left( 2 ^ { 2 } \times 5 \right) ^ { 2 }{/tex}
Multiplying with 3 we get
{tex}3 ^ { 2 } \times \left( 2 ^ { 2 } \times 5 \right) ^ { 2 } = ( 3 \times 4 \times 5 ) ^ { 2 } = 60 ^ { 2 }{/tex}
The required smallest natural number is 3.
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Yogita Ingle 7 years, 1 month ago
A collection of rational numbers and irrational numbers make up the set of real number. A real number can be expressed on the number line and has some specific properties. They satisfy:
- The commutative law of addition. That is, when a and b are two real numbers then a + b = b + a. For example 1 + 3 = 3 + 1 = 4
- The commutative law of multiplication. That is, when a and b are two real numbers then a x b = b x a. For example 1 x 3 = 3 x 1 = 3
- The associative law of addition. That is, when a, b and c are three real numbers then a + (b + c) = (a + b) + c. For example 1 + (3 + 4) = (1 + 3) + 4 = 8
- The associative law of multiplication. That is, when a, b and c are three real numbers then a x (b x c) = (a x b) x c. For example, 1 x (3 x 4) = (1 x 3) x 4 = 12
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