Problem 40 - Prealgebra

Question

The following problems review material from a previous section. Reviewing these problems will help you with the next section. Write as a decimal. $$\frac{120}{3}$$

Step-by-Step Solution

Answer

The fraction \( \frac{120}{3} \) as a decimal is 40.0.

Step 1: Identify the Division Operation

In this exercise, we are asked to convert a fraction to a decimal by performing division. The fraction given is \( \frac{120}{3} \). This means we need to divide 120 by 3.

Step 2: Perform the Division

To find the decimal form of \( \frac{120}{3} \), divide 120 by 3. Start by looking at how many times 3 goes into the first digit of 120, which is 1. It can't, so consider the first two digits, 12. 3 goes into 12 four times, i.e., 3 times 4 equals 12.

Step 3: Calculate the Full Quotient

Subtract 12 (from 3 times 4) from 12 in the dividend, and you'll get 0. Bring down the next digit, which is a 0, making it 00. 3 goes into 0 zero times. Thus, place a 0 above the line, next to the 4. The full quotient is 40.

Step 4: Write the Result as a Decimal

Since there’s no remainder and 3 divided 120 completely, the result is simply the quotient we obtained, which is 40. As a decimal, this is written as 40.0.

Key Concepts

Division Operation

When you're converting a fraction into a decimal, you're performing a key mathematical process called the division operation. The fraction consists of a numerator and a denominator, which in this case are 120 and 3, respectively. In the context of division, 120 is known as the dividend, and 3 is the divisor. The goal is to determine how many times the divisor (3) can go into the dividend (120), which ultimately gives us the decimal form of the fraction.

When starting to divide, consider each digit of the dividend sequentially:

• First, see how many times the divisor can fit into the first digit of the dividend.
• If it doesn't fit into the first digit alone, combine the next digits until the divisor fits.

The division operation is quite straightforward in this example, as shown in the solution. Hence, solving the fraction involves simplifying it by the number of times 3 fits into 120 without any remainder.

Quotients

The term 'quotient' is one you often hear in connection with division. The quotient is simply the result of the division operation. For any fraction or division problem, obtaining the quotient is the principal objective.

In our case, when we divide 120 by 3, we are seeking a number that represents how effectively 120 can be divided by 3. That's where our quotient, 40, comes into play:

• Divide 120 by 3, and you'll find that 3 fits perfectly into 120 exactly 40 times.
• This means that the quotient, which shows the exact answer to our division, is 40.
• The quotient is written as it is when there is no remainder in the division process.

A clear understanding of quotients allows you to transition comfortably to expressing fractions as decimals, and it clarifies how well the division process has been executed.

Fraction to Decimal Conversion

The journey from a fraction to a decimal involves using the division operation to derive the quotient. Here's a closer look at converting a fraction like \( \frac{120}{3} \) to a decimal:

Start by dividing the numerator by the denominator. If the division results in a whole number without any remainder, like our example, then you simply write down the quotient as your decimal.

• In this particular case, dividing 120 by 3 results in 40.
• Since there is no remainder, this division is complete.
• Thus, the decimal form of \( \frac{120}{3} \) is 40, or more precisely 40.0, indicating a whole number in decimal terms.

Understanding fraction to decimal conversions is crucial because it simplifies representation in various contexts, such as calculations requiring decimal precision or when performing further arithmetic operations involving decimals.