The numerator of a fraction is 8 less than the denominator. If the denominator is increased by 5 and the numerator is increased by 3, we get the fraction \(\tfrac34\). Find the original fraction.
\(\tfrac{27}{35}\)
Let the denominator be d, so the numerator is d-8.
New fraction: \(\dfrac{(d-8)+3}{d+5} = \dfrac34 \Rightarrow \dfrac{d-5}{d+5} = \dfrac34\).
\(4(d-5) = 3(d+5) \Rightarrow 4d-20 = 3d+15 \Rightarrow d = 35\).
Numerator: \(35-8 = 27\).
Hence, the original fraction is \(\tfrac{27}{35}\).
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Important Questions from Algebric Equations
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
Simplify the following expression:
\(\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x - y)(y - z)(z - x)}\)
