Problem 58 - Algebra 1: Concepts and Skills

Question

Write the fraction or mixed number as a decimal. (Skills Review pp. 763,767) $$ \frac{51}{12} $$

Step-by-Step Solution

Answer

The decimal representation of the fraction \(\frac{51}{12}\) is 4.25.

Step 1: Identify the Numerator and Denominator

The numerator of the fraction is 51 and the denominator is 12. We'll perform a division of these two numbers.

Step 2: Perform Division

Now, we proceed to perform the division operation. Therefore, 51 ÷ 12 = 4.25.

Step 3: Write Decimal

So, the fraction \(\frac{51}{12}\) can be written as the decimal 4.25.

Key Concepts

Numerator and Denominator

In any fraction, we have two key components: the numerator and the denominator. The numerator is the number on top. It represents the parts we have. For instance, if you slice a pizza into 8 parts and eat 3, your numerator is 3. Below the fraction line lies the denominator. The denominator tells us how many equal parts the whole was divided into. Using our pizza example, with a denominator of 8, we know the pizza was sliced into 8 equal pieces. In our exercise, the fraction \( \frac{51}{12} \) has 51 as the numerator and 12 as the denominator. Understanding these roles helps us perform division correctly.

Long Division

To convert a fraction to a decimal, we use a method called long division. Long division is where you divide the numerator by the denominator. It’s similar to basic division, but it might involve more steps. Let’s look at it step-by-step:

• Divide: Start by dividing the first part of the numerator by the denominator.
• Multiply: Multiply the divisor (denominator) by the result of the division and write it under the numerator.
• Subtract: Subtract the result from the numerator.
• Bring Down: If there’s more of the numerator remaining, bring down the next digit and repeat.

For example, dividing 51 by 12 gives us 4, since 12 fits 4 times into 51 without exceeding it. Multiply 12 by 4 to get 48, subtract from 51 to leave us with 3. Bring down a zero to make 30 and divide again to get further decimal places. That's how 51 ÷ 12 equals 4.25.

Writing Decimals

Writing decimals is about understanding the relationship between numbers. Decimals follow a base-10 system, often found after the decimal point. When we find 51 divided by 12 equals 4.25, the result tells us how many whole parts and fractional parts exist. The number 4 is the whole part or the number of complete sets we have. To the right of the decimal point, .25 tells us about the partial sets. It represents 25 out of 100 in simplified forms. Decimals are useful because they provide precision, showing even smaller portions than whole numbers or fractions can easily demonstrate.

Mixed Numbers

A mixed number combines a whole number and a fraction. For instance, \(4\frac{1}{4}\) is a mixed number where 4 is a whole and \(\frac{1}{4}\) is a fraction.When converting fractions like \(\frac{51}{12}\) into decimals, there's an intermediate step where the fraction can often be expressed as a mixed number. This is different from a simple fraction because it shows the whole, separate from the part. For example, \(\frac{51}{12}\) as a division results in 4 as the whole number, and after subtracting 48 (which is 12 times 4), you're left with \(\frac{3}{12}\). So, \(\frac{51}{12}\) can first be written as \(4\frac{3}{12}\), then simplified further to the decimal 4.25. Recognizing mixed numbers helps relate fractions and decimals to the whole parts and fractions separately.