Question

Verify and name the property p x (y+z) = p × y + p x z by taking p = -2/3, y = -4/5, z = -7/9.​

Answer 1

★ How to do :-

Here, we are given with three fractions in which we are asked to verify these three fractions in the form as given in the question. This property is also known as the distributive property. In some of the questions, we will have multiplication and addition, whereas in some of the questions, we'll have the multiplication and subtraction. In this question, we are given with the addition and multiplication. So, this can be considered as a distributive property. We should verify that the LHS and RHS of this question are equal or not. So, let's solve!!

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➤ Solution :-

${\sf \leadsto \boxed{\sf P \times (Y + Z)} = \boxed{\sf (P \times Y) + (P \times Z)}}$

Substitute the values on P, Y and Z.

${\tt \leadsto \dfrac{(-2)}{3} \times ( \dfrac{(-4)}{5} + \dfrac{(-7)}{9} = \dfrac{(-2)}{3} \times \dfrac{(-4)}{5} + \dfrac{(-2)}{3} \times \dfrac{(-7)}{9}}$

Let's solve the LHS and RHS separately.

LHS :-

${\tt \leadsto \dfrac{(-2)}{3} \times \bigg( \dfrac{(-4)}{5} + \dfrac{(-7)}{9} \bigg)}$

Solve the brackets first.

LCM of 5 and 9 is 45.

${\tt \leadsto \dfrac{(-2)}{3} \times \bigg( \dfrac{(-4) \times 9}{5 \times 9} + \dfrac{(-7) \times 5}{9 \times 5} \bigg)}$

Multiply the numerators and denominators of both fractions.

${\tt \leadsto \dfrac{(-2)}{3} \times \bigg( \dfrac{(-36)}{45} + \dfrac{(-35)}{45} \bigg)}$

Write both numerators with a common denominator.

${\tt \leadsto \dfrac{(-2)}{3} \times \bigg( \dfrac{(-36) + (-35)}{45} \bigg)}$

Write the second number in numerator with one sign.

${\tt \leadsto \dfrac{(-2)}{3} \times \bigg( \dfrac{(-36) - 35}{45} \bigg)}$

Subtract the numbers in the bracket.

${\tt \leadsto \dfrac{(-2)}{3} \times \dfrac{(-71)}{45}}$

Write both numerators and denominators in a common fraction.

${\tt \leadsto \dfrac{(-2) \times (-71)}{3 \times 45}}$

Multiply the numbers now.

${\tt \leadsto \dfrac{142}{135} \: --- \sf LHS}$

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Now, let's solve the RHS.

RHS :-

${\tt \leadsto \dfrac{(-2)}{3} \times \dfrac{(-4)}{5} + \dfrac{(-2)}{3} \times \dfrac{(-7)}{9}}$

Write the fractions to be multiplied in a common fraction.

${\tt \leadsto \dfrac{(-2) \times (-4)}{3 \times 5} + \dfrac{(-2) \times (-7)}{3 \times 9}}$

Multiply the numerators and denominators of both fractions.

${\tt \leadsto \dfrac{8}{15} + \dfrac{14}{27}}$

LCM of 15 and 27 is 135.

${\tt \leadsto \dfrac{8 \times 9}{15 \times 9} + \dfrac{14 \times 5}{27 \times 5}}$

Multiply the numerators and denominators of both fractions.

${\tt \leadsto \dfrac{72}{135} + \dfrac{70}{135}}$

Add those fractions now.

${\tt \leadsto \dfrac{72 + 70}{135} = \dfrac{142}{135} \: --- \sf RHS}$

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Now, let's compare the fractions.

${\tt \leadsto \dfrac{142}{135} \: \: and \: \: \dfrac{142}{135}}$

We can see that they are equal. So,

${\tt \leadsto \dfrac{142}{135} = \dfrac{142}{135}}$

So,

${\sf \leadsto LHS = RHS}$

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${\red{\underline{\boxed{\bf The \: property \: used \: here \: is \: Distributive \: property.}}}}$