Problem 47 - 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. $$\frac{46}{0.25}$$

Step-by-Step Solution

Answer

184

Step 1: Understanding Fractions as Division

The fraction \( \frac{46}{0.25} \) represents the division of 46 by 0.25. This means you need to calculate how many times 0.25 fits into 46.

Step 2: Simplify the Division

To make the division easier, multiply both the numerator (46) and the denominator (0.25) by 100 to get rid of the decimal. This changes the expression into \( \frac{4600}{25} \).

Step 3: Perform the Division

Now divide 4600 by 25. Determine how many times 25 goes into 4600 by performing long division or using a calculator. You find that 25 fits into 4600 exactly 184 times.

Step 4: Conclusion with the Decimal Form

Since 25 goes into 4600 exactly 184 times, the fraction \( \frac{46}{0.25} \) is equal to 184 when expressed as a decimal.

Key Concepts

Fractions in Division

When you see a fraction, it essentially signifies a division operation. A fraction like \( \frac{46}{0.25} \) means that 46 is divided by 0.25. Rather than merely a simple fraction, it demonstrates how one number can be divided or partitioned by another.

• Numerator: Represents the number to be divided (in this case, 46).
• Denominator: Represents the number you divide by (here, 0.25).

This division tells us how many groups of 0.25 are there in 46. Whenever you have a fraction, remember this key idea. It simplifies understanding and solving problems like converting fractions to decimals.

Simplifying Fractions

Simplifying fractions involves making the math more straightforward by eliminating any decimals from the divisor. By turning decimals into whole numbers, division becomes easier. One great strategy is to multiply both the numerator and the denominator by the same number to shift the decimal point.

Removing Decimals

Take \( \frac{46}{0.25} \). Here, multiplying 0.25 by 100 moves the decimal two places to the right, transforming it into a whole number. However, we must also multiply the numerator, 46, by the same 100. Thus, it becomes \( \frac{4600}{25} \). This approach to simplifying makes the division process smoother by converting it into long division with whole numbers.

Long Division

Long division is a reliable method to perform division with whole numbers. It's particularly useful in simplifying expressions or calculating exact results without a calculator.

• Start by determining how many times the divisor fits into the initial segment of the dividend.
• In our example, see how many times 25 fits into 46—2 times. Write 2 above the line and multiply, placing your result under 46, and subtract internally.
• Bring down the next digit from the dividend, repeating these steps until you finish.

With \( \frac{4600}{25} \), 25 fits into 4600 exactly 184 times without any remainder. This process, though initially complex, soon becomes your powerful tool for division and transformation from fractions to decimals.