Problem 12 - Prealgebra

Question

Find each of the following quotients. (Divide.) [Examples 1–5] $$-35 \div 7$$

Step-by-Step Solution

Answer

The quotient of -35 divided by 7 is -5.

Step 1: Understand the Division

To divide , you are essentially determining how many times 7 fits into -35, or what number multiplied by 7 results in -35.

Step 2: Perform the Division

Calculate the division: -35 ÷ 7 = -5. This tells us that 7 goes into -35 a total of -5 times.

Step 3: Verify the Result

To ensure our answer is correct, verify the result by multiplying the quotient back by the divisor: -5 × 7 = -35. The result matches the original dividend, confirming that our quotient is correct.

Key Concepts

Integer Division

When we talk about integer division, we are referring to the process of dividing one integer by another and finding a quotient that is also an integer. In the example \(-35 \div 7\), we wanted to see how many times 7 divides into -35. Importantly, when you perform integer division, the result is always an integer, meaning there is no remainder. If you have leftover from the division, it's ignored and only the whole number is considered as the quotient. For instance, dividing 10 by 3 in integer division gives you 3, not 3.333, since 3 fits into 10 exactly three times, with 1 remaining, which we ignore.

Negative Numbers

Negative numbers can seem tricky when performing operations like division, but the rules are actually straightforward. When you divide a positive integer by a negative one, or vice versa, the result is negative. This is why \(-35 \div 7 = -5\). When dividing two numbers of the same sign (both positive or both negative), the result is positive. Let's break it down further:

• Positive \( \div \) Positive = Positive
• Negative \( \div \) Negative = Positive
• Positive \( \div \) Negative = Negative
• Negative \( \div \) Positive = Negative

Remember, these rules apply to both multiplication and division, making it easier to learn and remember.

Multiplication Verification

Once you have your quotient from a division problem, checking your work with multiplication is a good practice. Multiplication verification helps confirm that your division was executed correctly. In our example, after determining that \(-35 \div 7 = -5\), you multiply the quotient back by the divisor to see if you get the initial dividend. Here’s what that looked like: \(-5 \times 7 = -35\)The multiplication shows that the division was correct because it yielded the original dividend. This verification step is especially useful as it allows you to confidently assert your answers without second-guessing. It's always a good idea to double-check mathematics operations, especially when negative numbers are involved.