Problem 52 - Prealgebra

Question

Choose the greatest number in each set. $$\left\\{\frac{3}{4}, 0.70,78 \%, 4 \text { out of } 5\right\\}$$

Step-by-Step Solution

Answer

The greatest number is 0.8 (from '4 out of 5').

Step 1: Convert Fractions to Decimals

First, convert all fractions into decimals. The fraction \(\frac{3}{4}\) can be converted into a decimal by dividing the numerator by the denominator: \(3 \div 4 = 0.75\). Next, the phrase '4 out of 5' is a fraction \(\frac{4}{5}\). Convert this to a decimal: \(4 \div 5 = 0.8\).

Step 2: Convert Percentages to Decimals

The percentage \(78\%\) can be converted to a decimal by dividing by 100: \(78\% = 0.78\). Now, all numbers in the set are in decimal form: \{0.75, 0.70, 0.78, 0.80\}.

Step 3: Compare All Decimals

Now that all the numbers are converted into decimal form, compare them. The decimals are 0.75, 0.70, 0.78, and 0.80. Among these, check which is the greatest. The greatest number is 0.80.

Key Concepts

Converting Fractions to Decimals

Converting fractions into decimals is an essential skill that helps simplify the process of comparing numbers. When you have a fraction like \(\frac{3}{4}\), you are essentially looking at a division operation, where the numerator, 3, is divided by the denominator, 4. When you perform this division, you get 0.75.

• Numerator: The top number in a fraction representing the number of parts.
• Denominator: The bottom number in a fraction showing the total number of equal parts.

Another example is the phrase '4 out of 5,' which represents the fraction \(\frac{4}{5}\). Dividing 4 by 5 gives you 0.8.
By converting fractions into decimals, you can perform straightforward comparisons with decimal numbers, making your calculations easier and quicker.

Converting Percentages to Decimals

Percentage is another form of expressing numbers that can easily be converted into decimals. A percentage such as 78% signifies 78 out of 100. To convert it into a decimal, you simply divide by 100. This process moves the decimal point two places to the left, resulting in 0.78.

• This conversion helps you compare percentages directly with decimals.
• It standardizes different number formats into a uniform form (decimals).

Once in decimal form, it becomes much simpler to rank or compare the values numerically. Understanding this conversion helps in various real-life applications, like calculating discounts or determining percentage changes.

Decimals Comparison

When all numbers are converted into decimals, you can easily compare them by looking at the digits from the left to right. The decimal with the greatest whole number or tenths decimal place is the largest.

• Starting with the whole part (or tenths in the absence of whole part): Compare the digits.
• If the digits in one place are the same, move to the next place to the right.

For instance, when you have numbers 0.75, 0.70, 0.78, and 0.80, comparing begins at the tenths place. Here, 0.8 contains "8" in the tenths place, which is higher than "7" and "5."
Always start your comparison with the most significant digit, which in decimals is typically the tenths digit, allowing you to determine magnitude with accuracy.