Question

Solve the following. Carry division to the hundredths place where necessary. \(5.7 \div 0.9=\) ______

Short answer

6.33

Step-by-step solution

Set up the Division Problem

To divide 5.7 by 0.9, first consider the division as a fraction: \( \frac{5.7}{0.9} \). The goal is to eliminate the decimal in the divisor (0.9).

Eliminate the Decimal in the Divisor

To eliminate the decimal in the divisor, multiply both the numerator and the denominator by 10. This step transforms \( \frac{5.7}{0.9} \) into \( \frac{57}{9} \).

Perform the Division

Now divide 57 by 9. Begin by seeing how many times 9 fits into 57. Calculate by finding the largest integer such that 9 times it is closest to (but not greater than) 57.

Long Division Calculation

9 goes into 57 six times since 9 \( \times \) 6 = 54. Subtract 54 from 57 to find a remainder of 3.

Continue to Hundredths Place

To carry the division to the hundredths place, add a decimal point and two zeroes to the remainder 3, making it 3.00. Bring down a zero. 9 goes into 30 three times (since 9 * 3 = 27). Subtract 27 from 30 to get 3.

Finalize Division

Bring down another zero to make it 30. Again, 9 goes into 30 three times. Subtract 27 to get 3. The decimal part stabilizes at 3, repeating.

Key concepts

Understanding Division Problems

In mathematics, division problems can often be compared to breaking a large pizza into smaller slices. Imagine you have a number of items or quantity and you want to distribute it evenly over some groups. This is the essence of division. When you see a division problem like "5.7 divided by 0.9," think of it as distributing 5.7 "units" equally into 0.9 "units" containers.

• Division problems can involve whole numbers or decimals.
• Integers or decimals can serve as both the dividend (the number to be divided) and the divisor (the number by which you divide).

The ultimate goal? To find out how many sets of the divisor fit into the dividend. When solving manual division problems, it's like being a detective, piecing together each part of the whole.

The Long Division Method

Long division is a classic pencil-and-paper approach to dividing numbers, especially useful for dividing larger numbers and decimals. It's a systematic method that breaks down a large division problem into smaller, easier steps.
To perform long division:

• First, write the dividend inside the division bracket and the divisor outside.
• Continue by determining how many times the divisor can fit into the first portion of the dividend.
• Place that number above the division bar, multiply it by the divisor, and subtract the result from the dividend.
• Bring down the next digit of the dividend and repeat the process until every digit has been accounted for.

Long division also handles decimals neatly. You simply need to keep track of the decimal point in your calculations and where it should ultimately be in the quotient.

Working with Decimal Division

Decimal division—dividing decimal numbers—adds a layer of complexity to basic division. Here's how it works:
When dividing by a decimal, first aim to transform the divisor into a whole number. This is achieved by multiplying the divisor by ten, or a suitable power of ten, shifting the decimal point to the right. To keep the equation balanced, do the same multiplication with the dividend.
For instance, in dividing 5.7 by 0.9, we multiply both by 10, effectively turning the question into 57 divided by 9.

• Position the decimal point in the recommended place in the quotient right above where it appears in the dividend.
• Continue with long division as you would with whole numbers.

The division carries on until you've reached your desired precision, or the results start repeating. This conversion method simplifies handling decimals, making the division straightforward.