Question

Find a rational number which when multipled by 4/3 and then subtracted 2/5 from the product gives 8/15​

Answer 1

$\huge{ \boxed{ \mathbb{ \red{Answer: }}}}$

$\implies{Number = -1/2}$

$\huge{ \underline{ \mathbb{ \purple{Solution:}}}}$

Let p/q (q ≠ 0) denote the rational number. Multiplying it by 5/2 gives (p/q)(5/2). Add 2/3 to the product and we get (p/q)(5/2) + 2/3 . The result is given to be -7/12.

∴ (p/q)(5/2) + 2/3 = -7/12 …………………………..……………………………………..(1)

Transposing 2/3 to right-hand-side and changing the sign to negative,

(p/q)(5/2) = -2/3 -7/12 = -(2/3 + 7/12) =- (2 x 4 + 7)/12 (Taking L.C.M.)

Or, (p/q)(5/2) = -(8+7)/12 = - 15/12

Multiplying both sides by 2/5,

(p/q)(5/2) x (2/5) = -15/12 . 2/5 = -(3x5)/(3x4) . 2/5 =- 5/4 .2/5

Since 2/5 is the multiplicative inverse of 5/2, 5/2 x 2/5 = 1 and we obtain

(p/q).1 = -1/4 . 2/1 = -1/2

⇒ p/q = -1/2 which is a negative rational number in which p = 1 and q = 2 ≠ 0 .

∴ the rational number = -1/2

$\underline{ \mathbb{ \purple{Check:}}}$

Substituting for p/q = -1/2 into the L.H.S. of (1),

(p/q)(5/2) + 2/3 = (-1/2)(5/2) + 2/3 = -5/4 + 2/3

= (-5 x 3 + 4 x 2)/(4.3) (Taking L.C.M.)

=(-15 + 8)/12 = -7/12 = R.H.S.

Answer 2

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Let p/q (q ≠ 0) denote the rational number. Multiplying it by 5/2 gives (p/q)(5/2). Add 2/3 to the product and we get (p/q)(5/2) + 2/3 . The result is given to be -7/12.

∴ (p/q)(5/2) + 2/3 = -7/12 …………………………..……………………………………..(1)

Transposing 2/3 to right-hand-side and changing the sign to negative,

(p/q)(5/2) = -2/3 -7/12 = -(2/3 + 7/12) =- (2 x 4 + 7)/12 (Taking L.C.M.)

Or, (p/q)(5/2) = -(8+7)/12 = - 15/12

Multiplying both sides by 2/5,

(p/q)(5/2) x (2/5) = -15/12 . 2/5 = -(3x5)/(3x4) . 2/5 =- 5/4 .2/5

Since 2/5 is the multiplicative inverse of 5/2, 5/2 x 2/5 = 1 and we obtain

(p/q).1 = -1/4 . 2/1 = -1/2

⇒ p/q = -1/2 which is a negative rational number in which p = 1 and q = 2 ≠ 0 .

∴ the rational number = -1/2

\underline{ \mathbb{ \purple{Check:}}}

Check:

Substituting for p/q = -1/2 into the L.H.S. of (1),

(p/q)(5/2) + 2/3 = (-1/2)(5/2) + 2/3 = -5/4 + 2/3

= (-5 x 3 + 4 x 2)/(4.3) (Taking L.C.M.)

=(-15 + 8)/12 = -7/12 = R.H.S.