Find using property -12/13 x 13/15 + 15/12 × 13/15

Let's evaluate the given expression:

\[
\frac{-12}{13} \times \frac{13}{15} + \frac{15}{12} \times \frac{13}{15}
\]

### Step 1: Simplify each product

1. \(\frac{-12}{13} \times \frac{13}{15}\):
   - Multiply the numerators and the denominators:
   \[
   \frac{-12 \times 13}{13 \times 15} = \frac{-156}{195}
   \]
   Since \(13\) appears in both the numerator and denominator, they cancel out:
   \[
   \frac{-12}{15}
   \]

2. \(\frac{15}{12} \times \frac{13}{15}\):
   - Multiply the numerators and the denominators:
   \[
   \frac{15 \times 13}{12 \times 15} = \frac{195}{180}
   \]
   Again, \(15\) appears in both the numerator and denominator, so they cancel out:
   \[
   \frac{13}{12}
   \]

### Step 2: Add the simplified results

Now, we need to add \(\frac{-12}{15}\) and \(\frac{13}{12}\).

To do that, we need a common denominator:
- The least common denominator (LCD) of \(15\) and \(12\) is \(60\).

Convert both fractions:
\[
\frac{-12}{15} = \frac{-48}{60}, \quad \frac{13}{12} = \frac{65}{60}
\]

### Step 3: Add the fractions
Now add:
\[
\frac{-48}{60} + \frac{65}{60} = \frac{-48 + 65}{60} = \frac{17}{60}
\]

Thus, the value of the expression is:

\[
\frac{17}{60}
\]

Would you like further details on this? 

Here are some related questions for practice:
1. Simplify \(\frac{7}{9} \times \frac{3}{4} + \frac{5}{6} \times \frac{7}{9}\).
2. Find the value of \(\frac{-5}{7} \times \frac{14}{9} + \frac{9}{14} \times \frac{14}{9}\).
3. Simplify \(\frac{3}{4} \times \frac{4}{7} + \frac{7}{3} \times \frac{4}{7}\).
4. Evaluate \(\frac{-8}{11} \times \frac{11}{9} + \frac{9}{8} \times \frac{11}{9}\).
5. Solve \(\frac{-10}{3} \times \frac{3}{4} + \frac{12}{5} \times \frac{3}{4}\).

**Tip**: Always reduce fractions at the earliest opportunity to make calculations simpler!

Learn how to simplify the expression -12/13 x 13/15 + 15/12 × 13/15 using the properties of fractions. This solution covers multiplication and addition of fractions with step-by-step guidance. Suitable for students in Grades 7-8.

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