if C equals three fractions, what is the value of C?

Question

$$ c = \frac{30}{7}\cdot \frac{999}{300} \cdot \frac{699}{100}$$ What is the value of c ? Is it smaller than 10? between 100 and 1000 ? between 10 and 100? or smaller than 1000?

How can i find that ?

Answer 1

$$\frac{30}{7}\cdot\frac{999}{300}\cdot\frac{699}{100}=\frac{999\cdot699}{7\cdot1000}=\frac{999}{1000}\cdot\frac{699}{7}<1\cdot100=100.$$ In another hand,

$$\frac{30}{7}\cdot\frac{999}{300}\cdot\frac{699}{100}=\frac{999\cdot699}{7\cdot1000}=\frac{999}{1000}\cdot\frac{699}{7}=\left(1-\frac{1}{1000}\right)\left(100-\frac{1}{7}\right)>$$ $$>100-\frac{1}{10}-\frac{1}{7}>99.$$