Arrange the following fractions in the descending order.
`10/12,13/15,21/25` and `43/45`

Answer

To arrange the fractions \( \frac{10}{12}, \frac{13}{15}, \frac{21}{25}, \frac{43}{45} \) in descending order, we will follow these steps:

### Step 1: Find the LCM of the Denominators
The denominators are 12, 15, 25, and 45. We will find the least common multiple (LCM) of these numbers.

- **Prime Factorization**:
- \( 12 = 2^2 \times 3^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 25 = 5^2 \)
- \( 45 = 3^2 \times 5^1 \)

- **LCM Calculation**:
- Take the highest power of each prime:
- \( 2^2 \) from 12
- \( 3^2 \) from 45
- \( 5^2 \) from 25
- Therefore, \( \text{LCM} = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900 \).

### Step 2: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of 900.

1. **For \( \frac{10}{12} \)**:
\[ \frac{10}{12} = \frac{10 \times 75}{12 \times 75} = \frac{750}{900} \]

2. **For \( \frac{13}{15} \)**:
\[ \frac{13}{15} = \frac{13 \times 60}{15 \times 60} = \frac{780}{900} \]

3. **For \( \frac{21}{25} \)**:
\[ \frac{21}{25} = \frac{21 \times 36}{25 \times 36} = \frac{756}{900} \]

4. **For \( \frac{43}{45} \)**:
\[ \frac{43}{45} = \frac{43 \times 20}{45 \times 20} = \frac{860}{900} \]

### Step 3: Compare the Numerators
Now we can compare the numerators of the fractions since they all have the same denominator:

- \( \frac{750}{900} \) (from \( \frac{10}{12} \))
- \( \frac{780}{900} \) (from \( \frac{13}{15} \))
- \( \frac{756}{900} \) (from \( \frac{21}{25} \))
- \( \frac{860}{900} \) (from \( \frac{43}{45} \))

### Step 4: Arrange in Descending Order
Now we can arrange the fractions based on their numerators:

1. \( \frac{43}{45} = \frac{860}{900} \) (largest)
2. \( \frac{13}{15} = \frac{780}{900} \) (second largest)
3. \( \frac{21}{25} = \frac{756}{900} \) (third largest)
4. \( \frac{10}{12} = \frac{750}{900} \) (smallest)

### Final Answer
The fractions in descending order are:
\[ \frac{43}{45}, \frac{13}{15}, \frac{21}{25}, \frac{10}{12} \]