Multiplying & Dividing Rational Numbers - Pre Algebra

1. The product of three rational numbers is $-\frac{12}{35}$. Two of the numbers are $\frac{2}{5}$ and $-\frac{3}{7}$. What is the third number?
   Answer: $2$ Let the third number be $x$. Then $\frac{2}{5} \times \left(-\frac{3}{7}\right) \times x = -\frac{12}{35}$. First: $\frac{2}{5} \times \left(-\frac{3}{7}\right) = -\frac{6}{35}$. So $-\frac{6}{35} \times x = -\frac{12}{35}$, which gives $x = \frac{-12/35}{-6/35} = \frac{-12}{35} \times \frac{35}{-6} = 2$. Choice B mishandles the double negative. Choice C uses wrong arithmetic. Choice D combines both errors.
2. Consider the expression $(-0.4) \times (-1.25) \div (-0.8)$. When computed from left to right, what is the final result?
   Answer: $-0.625$ Following order of operations left to right: $(-0.4) \times (-1.25) = 0.5$. Then $0.5 \div (-0.8) = -0.625$. The key is tracking signs carefully: negative times negative gives positive, then positive divided by negative gives negative. Choice B misses the final negative sign. Choice C makes an error in the initial multiplication. Choice D combines both sign errors.
3. A recipe calls for $\frac{3}{4}$ cup of flour, but Sarah only has $\frac{1}{6}$ cup. She decides to make a proportionally smaller batch by dividing all ingredients by the same factor. If she uses all her flour, by what factor should she divide the other ingredients?
   Answer: $4.5$ Sarah needs to find what factor to divide by, which means finding $\frac{3/4}{1/6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = 4.5$. This means the original recipe is 4.5 times larger than what she can make, so she divides all ingredients by 4.5. Choice A gives the ratio $\frac{1/6}{3/4}$ instead. Choice B is the same as A in different form. Choice D gives the reciprocal as an improper fraction.

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