Problem 42 - Prealgebra

Question

FIND THE ERROR Roberto and Daniel are finding \(\frac{9}{10}+\frac{7}{12} .\) Who is correct so far? Explain your reasoning. Roberto \(\frac{9}{10}+\frac{7}{12}=\frac{9}{10} \cdot \frac{12}{12}+\frac{7}{12} \cdot \frac{10}{10}\) Daniel \(\frac{9}{10}+\frac{7}{12}=\frac{9+7}{10+12}\)

Step-by-Step Solution

Answer

Roberto is correct because his approach correctly finds a common denominator for the addition of fractions.

Step 1: Understand how to add fractions

To add two fractions, they must have a common denominator. Roberto converts each fraction to have a common denominator, while Daniel combines numerators and denominators incorrectly.

Step 2: Analyze Roberto's approach

Roberto rewrites \ \(\frac{9}{10}+\frac{7}{12}=\frac{9}{10} \cdot \frac{12}{12}+\frac{7}{12} \cdot \frac{10}{10}\ \). This step is correct because he changes each fraction to have a common denominator of 120 by multiplying \ \(\frac{9}{10}\ \) by \ \(\frac{12}{12}\ \) and \ \(\frac{7}{12}\ \) by \ \(\frac{10}{10}\ \).

Step 3: Evaluate Daniel's approach

Daniel tries to add fractions by simply adding the numerators and denominators together to form \ \(\frac{9+7}{10+12} \ \). This operation is incorrect as it does not provide a valid method for adding fractions.

Step 4: Verify the proper method with calculation

Using Roberto's method, convert both fractions: \ \(\frac{9}{10}=\frac{9 \times 12}{120}=\frac{108}{120}\ \), and \ \(\frac{7}{12}=\frac{7 \times 10}{120}=\frac{70}{120}\ \). The sum is \ \(\frac{108}{120} + \frac{70}{120} = \frac{178}{120}\ \).

Key Concepts

Common Denominator

When adding fractions, it is crucial to have a common denominator. This is the number at the bottom of the fraction, representing how many equal parts the whole is divided into. Without a common denominator, you can't accurately add fractions.
A common denominator is typically the least common multiple (LCM) of the denominators of the fractions being added. In our example, to add \( \frac{9}{10} \) and \( \frac{7}{12} \), we identify 120 as the LCM of 10 and 12. This means we need to convert both fractions to have 120 as their new denominator before they can be added.
How do you find the LCM? List the multiples of each denominator, find the smallest multiple they share, and use it as the common denominator. This makes sure your fractions speak the same 'language' and can be easily combined.

Fraction Conversion

Fraction conversion is necessary when your fractions don't initially share a common denominator. Roberto's approach is spot on! You need to adjust each fraction so they have the same base to work with when adding. To do this, multiply both the numerator and the denominator of each fraction by a number that will help them reach the common denominator.

• For \( \frac{9}{10} \), multiply by \( \frac{12}{12} \), resulting in \( \frac{108}{120} \).
• For \( \frac{7}{12} \), multiply by \( \frac{10}{10} \), creating \( \frac{70}{120} \).

Notice that the fractions' values do not change; only their appearances do, allowing them to be added together without affecting the result. This conversion is critical to ensure the addition remains accurate.

Fraction Addition Error

Adding fractions might seem straightforward, but there is a common mistake many make, as shown in Daniel's approach. The error in fraction addition occurs when you try to add both numerators and denominators directly—which is incorrect. This process leads to results that do not reflect any real-world concept of fraction addition.
By adding numerators and denominators directly, like \( \frac{9+7}{10+12} \), you create a nonsensical fraction because the direct sum of 9 and 7 does not relate to joining fractions of the same whole. To fix this, always ensure you find a common denominator first, then add only the numerators.
Avoiding this error is simple if you convert fractions properly before adding. Stick to finding a common denominator first, just like Roberto did, to ensure error-free and accurate results.