स. Check that each of the following is true. $$ \begin{array}{lll} ext { (i) } \frac{1}{3}=\frac{1}{4}+\frac{1}{3 \cdot 4} & ext { (ii) } \frac{1}{4}=\frac{1}{5}+\frac{1}{4 \cdot 5} & ext { (iii) } \frac{1}{5}=\frac{1}{6}+\frac{1}{5 \cdot 6} \ ext { b. Based on the examples in (a), write } \frac{1}{n} ext { as a sum of two unit fractions; that is, as a sum of fractions with numerator } 1 ext {. } \ \begin{array}{l}\frac{1}{4}+\frac{1}{3 \cdot 4}=\frac{1}{4}+\square ext { (Type an integer or a simplified fraction.) } \ ext { a. Start by simplifying } \frac{1}{3 \cdot 4} \ ext { Now, write each fraction with a common denominator and add. Choose the correct answer below. } \ ext { A. } \frac{1}{4}+\frac{1}{12}=\frac{2}{16} \ ext { B. } \frac{1}{4}+\frac{1}{12}=\frac{6}{12} \ ext { C. } \frac{1}{4}+\frac{1}{12}=\frac{16}{48}\end{array} \ ext { (i) } \ ext { (i) } \ ext { (i) }\end{array} $$

Question

स. Check that each of the following is true. $$ \begin{array}{lll}	ext { (i) } \frac{1}{3}=\frac{1}{4}+\frac{1}{3 \cdot 4} & 	ext { (ii) } \frac{1}{4}=\frac{1}{5}+\frac{1}{4 \cdot 5} & 	ext { (iii) } \frac{1}{5}=\frac{1}{6}+\frac{1}{5 \cdot 6} \ 	ext { b. Based on the examples in (a), write } \frac{1}{n} 	ext { as a sum of two unit fractions; that is, as a sum of fractions with numerator } 1 	ext {. } \ \begin{array}{l}\frac{1}{4}+\frac{1}{3 \cdot 4}=\frac{1}{4}+\square 	ext { (Type an integer or a simplified fraction.) } \ 	ext { a. Start by simplifying } \frac{1}{3 \cdot 4} \ 	ext { Now, write each fraction with a common denominator and add. Choose the correct answer below. } \ 	ext { A. } \frac{1}{4}+\frac{1}{12}=\frac{2}{16} \ 	ext { B. } \frac{1}{4}+\frac{1}{12}=\frac{6}{12} \ 	ext { C. } \frac{1}{4}+\frac{1}{12}=\frac{16}{48}\end{array} \ 	ext { (i) } \ 	ext { (i) } \ 	ext { (i) }\end{array} $$

UpStudy AI · Accepted Answer

Let's check each of the statements in part (a) one by one.

### Part (a)

1. **Check (i)**: $$ \frac{1}{3} = \frac{1}{4} + \frac{1}{3 \cdot 4} $$

First, we need to simplify $$ \frac{1}{3 \cdot 4} $$:
   $$
   \frac{1}{3 \cdot 4} = \frac{1}{12}
   $$
   Now, we can rewrite the equation:
   $$
   \frac{1}{3} = \frac{1}{4} + \frac{1}{12}
   $$
   To add $$ \frac{1}{4} $$ and $$ \frac{1}{12} $$, we need a common denominator. The least common multiple of 4 and 12 is 12. Thus:
   $$
   \frac{1}{4} = \frac{3}{12}
   $$
   Now we can add:
   $$
   \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}
   $$
   So, statement (i) is true.

2. **Check (ii)**: $$ \frac{1}{4} = \frac{1}{5} + \frac{1}{4 \cdot 5} $$

First, simplify $$ \frac{1}{4 \cdot 5} $$:
   $$
   \frac{1}{4 \cdot 5} = \frac{1}{20}
   $$
   Now, rewrite the equation:
   $$
   \frac{1}{4} = \frac{1}{5} + \frac{1}{20}
   $$
   The least common multiple of 5 and 20 is 20. Thus:
   $$
   \frac{1}{5} = \frac{4}{20}
   $$
   Now we can add:
   $$
   \frac{4}{20} + \frac{1}{20} = \frac{5}{20} = \frac{1}{4}
   $$
   So, statement (ii) is true.

3. **Check (iii)**: $$ \frac{1}{5} = \frac{1}{6} + \frac{1}{5 \cdot 6} $$

First, simplify $$ \frac{1}{5 \cdot 6} $$:
   $$
   \frac{1}{5 \cdot 6} = \frac{1}{30}
   $$
   Now, rewrite the equation:
   $$
   \frac{1}{5} = \frac{1}{6} + \frac{1}{30}
   $$
   The least common multiple of 6 and 30 is 30. Thus:
   $$
   \frac{1}{6} = \frac{5}{30}
   $$
   Now we can add:
   $$
   \frac{5}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}
   $$
   So, statement (iii) is true.

### Part (b)

Based on the examples in (a), we can write $$ \frac{1}{n} $$ as a sum of two unit fractions. The pattern observed is:
$$
\frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)}
$$

For $$ n = 4 $$:
$$
\frac{1}{4} = \frac{1}{5} + \frac{1}{20}
$$

Now, let's fill in the blank:
$$
\frac{1}{4} + \frac{1}{3 \cdot 4} = \frac{1}{4} + \frac{1}{12}
$$

### Simplifying $$ \frac{1}{4} + \frac{1}{12} $$

To add these fractions, we need a common denominator. The least common multiple of 4 and 12 is 12:
$$
\frac{1}{4} = \frac{3}{12}
$$
Now we can add:
$$
\frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}
$$

### Choose the correct answer

The correct answer is:
**B.** $$ \frac{1}{4} + \frac{1}{12} = \frac{6}{12} $$
All statements (i), (ii), and (iii) are true. For $$ \frac{1}{n} $$, it can be expressed as $$ \frac{1}{n+1} + \frac{1}{n(n+1)} $$. For $$ \frac{1}{4} + \frac{1}{12} $$, the correct simplification is $$ \frac{6}{12} $$.

स. Check that each of the following is true.

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