Question

multiply:
$\frac{5}{10} \cdot (-3) \cdot \frac{7}{-6}$
write your answer in simplest form.

Explanation:

Step1: Multiply the numerators and denominators

We have the expression $\frac{5}{10} \cdot (-3) \cdot \frac{7}{-6}$. First, multiply the numerators: $5\times(-3)\times7 = -105$. Then multiply the denominators: $10\times1\times(-6)= -60$. So the fraction becomes $\frac{-105}{-60}$.

Step2: Simplify the fraction

A negative divided by a negative is positive, so $\frac{-105}{-60}=\frac{105}{60}$. Now, find the greatest common divisor (GCD) of 105 and 60, which is 15. Divide both the numerator and denominator by 15: $\frac{105\div15}{60\div15}=\frac{7}{4}$. Wait, no, wait, let's re - check the multiplication. Wait, the original expression is $\frac{5}{10}\cdot(-3)\cdot\frac{7}{-6}$. Let's rewrite $-3$ as $\frac{-3}{1}$. So we have $\frac{5}{10}\times\frac{-3}{1}\times\frac{7}{-6}$. Now, multiply numerators: $5\times(-3)\times7=-105$. Multiply denominators: $10\times1\times(-6) = - 60$. Then $\frac{-105}{-60}=\frac{105}{60}$. Now, simplify $\frac{105}{60}$: divide numerator and denominator by 15, we get $\frac{7}{4}$? Wait, no, 105÷15 = 7, 60÷15 = 4. But wait, maybe we can simplify earlier. Let's cancel out the signs first: the two negative signs (from - 3 and - 6) will make a positive. So we have $\frac{5}{10}\times3\times\frac{7}{6}$. Now, simplify $\frac{5}{10}$ to $\frac{1}{2}$. So now we have $\frac{1}{2}\times3\times\frac{7}{6}$. Multiply $\frac{1}{2}$ and 3: $\frac{3}{2}$. Then multiply $\frac{3}{2}$ and $\frac{7}{6}$: $\frac{3\times7}{2\times6}=\frac{21}{12}$. Simplify $\frac{21}{12}$ by dividing numerator and denominator by 3: $\frac{7}{4}$. Wait, but let's do it step by step correctly.

Alternative Step1: Simplify fractions first. $\frac{5}{10}=\frac{1}{2}$. The expression becomes $\frac{1}{2}\times(-3)\times\frac{7}{-6}$.
Alternative Step2: Deal with the signs. The product of two negative numbers (-3 and -6) is positive, so we can rewrite the expression as $\frac{1}{2}\times3\times\frac{7}{6}$.
Alternative Step3: Multiply the fractions. $\frac{1\times3\times7}{2\times1\times6}=\frac{21}{12}$.
Alternative Step4: Simplify $\frac{21}{12}$ by dividing numerator and denominator by 3: $\frac{21\div3}{12\div3}=\frac{7}{4}$. Wait, but let's check again. Wait, $\frac{5}{10}\times(-3)\times\frac{7}{-6}$. Let's compute the product of the fractions first. $\frac{5}{10}\times\frac{7}{-6}=\frac{5\times7}{10\times(-6)}=\frac{35}{-60}=-\frac{35}{60}=-\frac{7}{12}$. Then multiply by - 3: $-\frac{7}{12}\times(-3)=\frac{21}{12}=\frac{7}{4}$. Yes, that's correct.

Answer:

$\frac{7}{4}$