Solve 7.8/13<0.078/0.1378/0.13<0.0078/0.0000013

$$\frac{ 7.8 }{ 13 } < \frac{ 0.078 }{ 0.13 } \frac{ 78 }{ 0.13 } < \frac{ 0.0078 }{ 0.0000013 }$$

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Solution Steps

Expand \frac{7.8}{13} by multiplying both numerator and the denominator by 10.

$$\frac{78}{130}<\frac{0.078}{0.13}\times \frac{78}{0.13}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Reduce the fraction \frac{78}{130} to lowest terms by extracting and canceling out 26.

$$\frac{3}{5}<\frac{0.078}{0.13}\times \frac{78}{0.13}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Expand \frac{0.078}{0.13} by multiplying both numerator and the denominator by 1000.

$$\frac{3}{5}<\frac{78}{130}\times \frac{78}{0.13}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Reduce the fraction \frac{78}{130} to lowest terms by extracting and canceling out 26.

$$\frac{3}{5}<\frac{3}{5}\times \frac{78}{0.13}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Expand \frac{78}{0.13} by multiplying both numerator and the denominator by 100.

$$\frac{3}{5}<\frac{3}{5}\times \frac{7800}{13}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Divide 7800 by 13 to get 600.

$$\frac{3}{5}<\frac{3}{5}\times 600\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Express \frac{3}{5}\times 600 as a single fraction.

$$\frac{3}{5}<\frac{3\times 600}{5}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Multiply 3 and 600 to get 1800.

$$\frac{3}{5}<\frac{1800}{5}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Compare \frac{3}{5} and \frac{1800}{5}.

$$\text{true}\text{ and }\frac{0.078}{0.13}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Expand \frac{0.078}{0.13} by multiplying both numerator and the denominator by 1000.

$$\text{true}\text{ and }\frac{78}{130}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Reduce the fraction \frac{78}{130} to lowest terms by extracting and canceling out 26.

$$\text{true}\text{ and }\frac{3}{5}\times \frac{78}{0.13}<\frac{0.0078}{0.0000013}$$

Expand \frac{78}{0.13} by multiplying both numerator and the denominator by 100.

$$\text{true}\text{ and }\frac{3}{5}\times \frac{7800}{13}<\frac{0.0078}{0.0000013}$$

Divide 7800 by 13 to get 600.

$$\text{true}\text{ and }\frac{3}{5}\times 600<\frac{0.0078}{0.0000013}$$

Express \frac{3}{5}\times 600 as a single fraction.

$$\text{true}\text{ and }\frac{3\times 600}{5}<\frac{0.0078}{0.0000013}$$

Multiply 3 and 600 to get 1800.

$$\text{true}\text{ and }\frac{1800}{5}<\frac{0.0078}{0.0000013}$$

Divide 1800 by 5 to get 360.

$$\text{true}\text{ and }360<\frac{0.0078}{0.0000013}$$

Expand \frac{0.0078}{0.0000013} by multiplying both numerator and the denominator by 10000000.

$$\text{true}\text{ and }360<\frac{78000}{13}$$

Divide 78000 by 13 to get 6000.

$$\text{true}\text{ and }360<6000$$

Compare 360 and 6000.

$$\text{true}\text{ and }\text{true}$$

The conjunction of \text{true} and \text{true} is \text{true}.

$$\text{true}$$