High School

Simplify the following expression:

[tex]\[103 \frac{1}{2} \times 197 \frac{1}{2} \times 48\][/tex]

Answer :

We begin by converting the mixed numbers to improper fractions. The mixed number

[tex]$$103\frac{1}{2}$$[/tex]

can be written as

[tex]$$103\frac{1}{2} = 103 + \frac{1}{2} = \frac{103 \times 2 + 1}{2} = \frac{207}{2}.$$[/tex]

Similarly, the mixed number

[tex]$$197\frac{1}{2}$$[/tex]

becomes

[tex]$$197\frac{1}{2} = 197 + \frac{1}{2} = \frac{197 \times 2 + 1}{2} = \frac{395}{2}.$$[/tex]

Next, we multiply these two numbers:

[tex]$$\frac{207}{2} \times \frac{395}{2}.$$[/tex]

If we were to convert these fractions to decimal form, we have

[tex]$$\frac{207}{2} = 103.5 \quad \text{and} \quad \frac{395}{2} = 197.5.$$[/tex]

Multiplying the two decimals gives:

[tex]$$103.5 \times 197.5 = 20441.25.$$[/tex]

Finally, we multiply this product by 48:

[tex]$$20441.25 \times 48 = 981180.$$[/tex]

Thus, the final answer is

[tex]$$\boxed{981180}.$$[/tex]