Question

Solve the following. Carry division to the hundredths place where necessary. \(1.125 \div 0.75=\) ______

Short answer

The result is 1.5.

Step-by-step solution

Convert Division to a Simpler Equivalent

First, convert the division problem to an easier form by eliminating decimals. Multiply both the dividend (1.125) and the divisor (0.75) by 100 to make them whole numbers. So the new problem is: 112.5 divided by 75.

Simplify the Problem

To simplify dividing by 75, we can double the numbers (multiply by 2) to get easier whole numbers. Multiply both 112.5 and 75 by 2: 112.5 × 2 = 225, 75 × 2 = 150. Now solve 225 divided by 150.

Perform the Division

Next, perform the division of 225 by 150 using long division. - 150 goes into 225 once (1). - 225 - 150 = 75. Bring down a zero to make it 750, but since there are no more digits to bring down from 225, we directly consider the remainder.

Convert Remainder to Decimal

Consider 750, the digit you obtain by bringing down a zero should have been an extension from the decimal. Divide 750 by 150 to get 5. Hence, the decimal comes from 1 and a remainder of 0.5, making the full calculation 1.5.

Key concepts

Long Division

Long division is a method used for dividing larger numbers that cannot be easily divided mentally. It involves several steps, including dividing, multiplying, and subtracting, repeated in sequence. Here's how it works:

• Divide the first few digits of the dividend by the divisor.
• Multiply the quotient by the divisor, and write this number under the dividend.
• Subtract the result from the dividend or the part of the dividend used.
• Bring down the next digit of the dividend, and repeat the process.

Long division can initially seem complicated, but breaking it down into manageable steps makes the process clearer. This is especially helpful when dividing by numbers like 75, as in our example, where estimating how many times the divisor fits into the dividend is part of the process.

Decimal Division

Decimal division involves dividing numbers that have decimals. A common way to make the division of decimal numbers easier is to eliminate decimals:

• Move the decimal point to the right in both the dividend and divisor until both numbers become whole numbers.
• Perform the division using the long division method.
• If needed, adjust the final answer by placing the decimal point in the correct position.

In the original exercise, dividing 1.125 by 0.75 required transforming the numbers by multiplying both by 100 to get whole numbers: 112.5 and 75. After this adjustment, the operation becomes straightforward using long division.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible by reducing it to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.

• Check if both numbers can be divided by any small prime numbers (e.g., 2, 3, 5).
• Divide the numerator and denominator by their GCD.
• Keep simplifying until you cannot divide anymore without a remainder.

In decimal division, when you adjust the operation to whole numbers, you might end up with an improper fraction, like 225/150, which can be simplified to 1.5 by understanding that 150 divides into 225 completely with a remainder.

Multiplication in Division Process

In division, multiplication plays an essential role. It helps to generate new numbers to continue the division process and simplify the problem. Here's how:

• When eliminating decimals, multiplication helps create whole numbers, simplifying lining up numbers in long division.
• If the numbers involved in the division are difficult to estimate, sometimes multiplying both the dividend and divisor by the same number makes them easier to handle.
• The multiplication step in long division, where the current quotient is multiplied by the divisor, checks and adjusts the remainder.

For instance, multiplying both 112.5 and 75 by 2 simplifies the division process to deal with numbers like 225 and 150. This approach is handy in scenarios where both numbers end with multiplying or dividing steps.