TAOCP 4.5.1 Exercise 5
**Corrected Solution for Exercise 4.
Exercise 5. [**] [$10$] Compute $(17/120) \div (-27/70)$ by the method recommended in the text.
Verified: yes
Solve time: 4m15s
Corrected Solution for Exercise 4.5.1.5
We are asked to compute
$$ \frac{17}{120} \div \frac{-27}{70} $$
by the method recommended in the text (Section 4.5.1), which prescribes cross-cancellation before multiplication to keep intermediate numbers small.
Step 1: Replace division by multiplication with reciprocal
By definition,
$$ \frac{17}{120} \div \frac{-27}{70} = \frac{17}{120} \cdot \frac{70}{-27}. $$
We will now compute the product using cross-cancellation.
Step 2: Identify gcds for cross-cancellation
Cross-cancellation looks for common factors between numerators and denominators of opposite fractions.
- Between $17$ (numerator of the first fraction) and $27$ (denominator of the second fraction):
$$ \gcd(17,27) = 1 $$
No cancellation occurs here.
- Between $70$ (numerator of the second fraction) and $120$ (denominator of the first fraction):
$$ \gcd(70,120) = 10 $$
We divide both by $10$:
$$ 70/10 = 7, \quad 120/10 = 12 $$
So after cross-cancellation, the fractions become:
$$ \frac{17}{12} \cdot \frac{7}{-27}. $$
Step 3: Multiply numerators and denominators
Now we multiply the simplified numerators and denominators:
$$ \frac{17 \cdot 7}{12 \cdot (-27)} = \frac{119}{-324} = -\frac{119}{324}. $$
Step 4: Verify that the result is in lowest terms
The numerator $119 = 7 \cdot 17$ and the denominator $324 = 2^2 \cdot 3^4$ share no common factors. Hence the fraction is fully reduced.
Step 5: State the final answer
$$ \boxed{-\frac{119}{324}} $$
This computation follows the method recommended in the text, performing cross-cancellation using gcds before multiplying.
Answer:
$$ \displaystyle -\frac{119}{324} $$