Question

Find the quotient. $$-12.6 \div 1.8$$

Short answer

\(-7.00\)

Step-by-step solution

Write Division as Fraction

We can write this division problem as a fraction, which might be easier to solve. It will look like this: \(-12.6 / 1.8\).

Divide the Numbers

Next, simply perform the division operation. Divide -12.6 by 1.8.

Determine the Quotient

After performing the division and rounding to the nearest hundredth if necessary, you should get your answer. Remember, when dividing a negative number by a positive number, the quotient is negative.

Key concepts

Division Operation

Understanding the division operation is essential for solving problems requiring you to find the quotient of two numbers. Division can be seen as the process of determining how many times one number (the divisor) is contained within another (the dividend).

Let's apply this to our exercise, where we have \( -12.6 \div 1.8 \). The division operation here is between a negative decimal number and a positive decimal number. When dividing decimals, the goal is to determine how many times 1.8 fits into -12.6. One way to simplify this is to eliminate the decimal by multiplying both numbers by the same power of 10, which, in this case, is 10, so you're effectively calculating \( -126 \div 18 \).

Once this multiplication is done, you proceed to divide as you would with whole numbers. Remember to account for the decimal place you have shifted, to ensure your final answer is accurate. This method of multiplying both numbers to remove the decimal can greatly simplify the division process, especially when dealing with decimals that have multiple digits after the decimal point.

Negative Numbers Division

The division of negative numbers follows certain rules that are consistent with the principles of arithmetic. When dividing a negative number by a positive one, or vice versa, the result will always be a negative number. This is because the signs of the numbers are different.

In the given exercise, \( -12.6 \div 1.8 \) involves the division of a negative by a positive number. Hence, we know that our quotient will be a negative value. Understanding this rule helps avoid confusion and ensures accuracy in calculations involving negative numbers. It's a foundational concept in working with integers, and keeping the sign rules straight is crucial for reaching the correct solution in algebraic problems.

Additionally, when you divide two negative numbers, the result will be positive, since the negative signs cancel out. These sign rules are fundamental and apply no matter how complex the division operation is.

Fractions and Division

Division problems can often be rewritten as fractions for a more intuitive understanding and calculation. A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. The division line in a fraction represents the division between these two numbers.

For the exercise \( -12.6 \div 1.8 \) or, written as a fraction, \( -12.6 / 1.8 \), you treat the numerator \( -12.6 \) as the dividend and the denominator \( 1.8 \) as the divisor. Solving division by rewriting it as a fraction can also make the concept of inverting and multiplying clear—this is another way to divide fractions where you multiply by the reciprocal of the divisor.

Understanding the relationship between fractions and division enhances your mathematical literacy, making it easier to tackle problems involving ratios, rates, and proportions, which can often be expressed as fractions.