Problem 202 - 1,001 GED Practice Questions For Dummies

Question

\(3 \frac{1}{10}+1 \frac{11}{100}+2 \frac{111}{1,000}=\) (Note: You may not use a calculator.) (A) \(6.111\) (B) \(6.123\) (C) \(6.321\) (D) \(6.11111\)

Step-by-Step Solution

Answer

(C) \(6.321\)

Step 1: Convert mixed numbers to improper fractions

To convert mixed numbers to improper fractions, multiply the whole number by the denominator and add the numerator. This becomes the new numerator, and the denominator stays the same. 1. \(3 \frac{1}{10}\): Multiply 3 by 10 and add 1, which gives us \(3 \cdot 10 + 1 = 31\), so the fraction is \(\frac{31}{10}\). 2. \(1 \frac{11}{100}\): Multiply 1 by 100 and add 11, which gives us \(1 \cdot 100 + 11 = 111\), so the fraction is \(\frac{111}{100}\). 3. \(2 \frac{111}{1000}\): Multiply 2 by 1000 and add 111, which gives us \(2 \cdot 1000 + 111 = 2111\), so the fraction is \(\frac{2111}{1000}\).

Step 2: Find a common denominator

To find a common denominator for the three fractions, we need to find the least common multiple (LCM) of their denominators (10, 100, and 1000). The LCM of 10, 100, and 1000 is 1000.

Step 3: Convert fractions to the common denominator

Now, we need to convert each fraction to an equivalent fraction with the common denominator of 1000. 1. \(\frac{31}{10} = \frac{31 \cdot 100}{10 \cdot 100} = \frac{3100}{1000}\) 2. \(\frac{111}{100} = \frac{111 \cdot 10}{100 \cdot 10} = \frac{1110}{1000}\) 3. \(\frac{2111}{1000}\) already has the correct denominator.

Step 4: Add the fractions

Add the numerators and keep the common denominator of 1000. \(\frac{3100}{1000} + \frac{1110}{1000} + \frac{2111}{1000} = \frac{3100 + 1110 + 2111}{1000} = \frac{6321}{1000}\)

Step 5: Convert back to a mixed number and simplify

Divide the numerator (6321) by the denominator (1000) to get the whole number part and the remainder: \(6321 \div 1000 = 6 \text{ with a remainder of } 321\) So, our mixed number is \(6 \frac{321}{1000}\). Now, we compare the mixed number with the given options: (A) \(6.111\) (B) \(6.123\) (C) \(6.321\) (D) \(6.11111\) Our mixed number \(6 \frac{321}{1000}\) is equivalent to option (C) \(6.321\). Therefore, the answer is (C) \(6.321\).

Key Concepts

Fractions

Fractions represent parts of a whole. They are written with two numbers, one above the other, separated by a line. The top number is called the numerator, which tells how many parts we have. The bottom number is the denominator, showing into how many equal parts the whole is divided.

• The fraction \(\frac{1}{10}\) is read as "one-tenth," meaning one part out of ten equal parts.
• Fractions can also be larger than one, called improper fractions, where the numerator is larger than the denominator. For example, \(\frac{7}{3}\).

Understanding fractions is essential in dealing with mixed numbers, as they combine whole numbers and fractions.

Mixed Numbers

Mixed numbers contain both a whole number and a fraction. For example, in the mixed number \(3 \frac{1}{10}\), the "3" is the whole number, and "\(\frac{1}{10}\)" is the fraction part.

• To perform operations like addition, it's often easier to convert mixed numbers into improper fractions first.
• Multiply the whole number by the denominator, then add the numerator.

Using our example of \(3 \frac{1}{10}\), we convert it to \(\frac{31}{10}\). Doing so simplifies arithmetic operations and allows us to work with consistent forms.

Decimal Conversion

Decimals and fractions are two different ways of expressing the same concept. Converting between them can make certain calculations easier or more intuitive.

• To convert a fraction to a decimal, divide the numerator by the denominator.
• For example, \(\frac{1}{4}\) converts to 0.25.

With mixed numbers, the whole number remains as it is, and only the fractional part is converted to a decimal. So, \(3 \frac{1}{10}\) becomes 3.1 after conversion. Understanding these conversions aids in comprehension and comparison of different numerical forms.

Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of them. Finding the LCM is especially useful when adding or subtracting fractions because it helps us find a common denominator.

• For denominators like 10, 100, and 1000, the LCM is 1000.
• This means each fraction can be expressed with 1000 as the denominator without changing their value.

Once all fractions have the common denominator, simply add or subtract the numerators. This technique ensures seamless arithmetic operations across different fractions.