Understanding Negative Fractions in Mathematics

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Questions and Answers

Explain why the product of two negative decimals is positive, while the product of a positive and a negative decimal is negative.

Multiplying two negative numbers results in a positive number because the negative signs effectively cancel each other out. Multiplying a positive and a negative number results in a negative number because there is only one negative sign, so the result remains negative.

How does converting a negative fraction to a decimal affect its value? Provide an example.

Converting a negative fraction to a decimal does not change its value, only its representation. For example, $-3/4$ is equal to $-0.75$.

Solve: $\frac{-5}{8} / \frac{1}{4} = $?

$-\frac{5}{2}$

Explain how subtracting a negative decimal from another negative decimal affects the result. Give an example.

Subtracting a negative decimal is the same as adding a positive decimal. This effectively increases the value of the original negative decimal, moving it closer to zero. For example, $-0.75 - (-0.25) = -0.75 + 0.25 = -0.50$. Signup and view all the answers

Convert $-0.625$ to a simplified fraction.

$-\frac{5}{8}$ Signup and view all the answers

If $x = -0.4$ and $y = 0.8$, what is the value of $x - y$?

$-1.2$ Signup and view all the answers

Describe the difference between adding two negative decimals and adding a positive and a negative decimal.

Adding two negative decimals involves summing their absolute values and keeping the negative sign. Adding a positive and a negative decimal involves subtracting the smaller absolute value from the larger and using the sign of the number with the larger absolute value. Signup and view all the answers

True or false: A negative fraction divided by a negative fraction is always a positive number?

True Signup and view all the answers

Explain why $-\frac{3}{4}$ and $\frac{-3}{4}$ represent the same value. Provide a brief, intuitive explanation.

Both notations represent the same quantity because a negative sign applied to the entire fraction is equivalent to applying it to either the numerator or denominator, but not both. The value is still three-quarters less than zero. Signup and view all the answers

Describe the potential pitfall in computing $-\frac{1}{2} - \frac{3}{4}$ without finding a common denominator first. What is the correct answer?

Without a common denominator, one might incorrectly subtract the numerators directly or make an error in sign. Correct answer: $-\frac{5}{4}$ Signup and view all the answers

Explain the rule for multiplying two negative fractions, such as $(-\frac{2}{5}) \times (-\frac{1}{3})$, and provide the result.

When multiplying two negative fractions, the product is positive. Multiply numerators and denominators separately: $(-\frac{2}{5}) \times (-\frac{1}{3}) = \frac{2}{15}$. Signup and view all the answers

Why is it necessary to invert the second fraction when dividing by a fraction? Give an example using negative fractions. For example: $(-\frac{1}{2}) \div (\frac{3}{4})$

Inverting the second fraction and multiplying is equivalent to finding how many times the second fraction fits into the first, which is the definition of division. For the example: $(-\frac{1}{2}) \div (\frac{3}{4}) = (-\frac{1}{2}) \times (\frac{4}{3}) = -\frac{4}{6} = -\frac{2}{3}$. Signup and view all the answers

Consider the expression $-\frac{5}{8} + \frac{1}{4} - \frac{3}{2}$. Outline the steps to simplify it, ensuring you address how to handle the negative signs correctly.

First, find a common denominator: 8. Then, rewrite the expression as $-\frac{5}{8} + \frac{2}{8} - \frac{12}{8}$. Next, combine the numerators: $\frac{-5 + 2 - 12}{8} = \frac{-15}{8}$. The simplified result is $-\frac{15}{8}$. Signup and view all the answers

Provide a real-world example where negative fractions might be used, outlining what the numerator and denominator could represent in that scenario.

A real-world example is owing money. If you owe $\frac{1}{2}$ of a dollar, it can be represented as $-\frac{1}{2}$. The numerator (1) represents the amount owed, and the denominator (2) represents the total parts the dollar is divided into. Signup and view all the answers

Given the expression $(-\frac{1}{3}) \times (\frac{3}{5}) \div (-\frac{2}{3})$, what is the correct sequence of operations and the final simplified result?

First, perform the multiplication: $(-\frac{1}{3}) \times (\frac{3}{5}) = -\frac{3}{15} = -\frac{1}{5}$. Next, perform the division by multiplying by the reciprocal: $(-\frac{1}{5}) \div (-\frac{2}{3}) = (-\frac{1}{5}) \times (-\frac{3}{2}) = \frac{3}{10}$. The final simplified result is $\frac{3}{10}$. Signup and view all the answers

Explain the difference in the outcome between $-\frac{1}{2} + \frac{1}{4}$ and $-\frac{1}{2} \times \frac{1}{4}$.

For $-\frac{1}{2} + \frac{1}{4}$, we add the fractions, resulting in $-\frac{2}{4} + \frac{1}{4} = -\frac{1}{4}$. For $-\frac{1}{2} \times \frac{1}{4}$, we multiply the fractions, resulting in $-\frac{1}{8}$. Addition combines the quantities with like signs, while multiplication scales one fraction by another. Signup and view all the answers

Flashcards

Negative Decimal

A decimal less than zero, indicated by a minus sign.

Adding Negative Decimals (both negative)

Add absolute values and keep the negative sign.