How do I add decimal expansions without a calculator

Question

Really not sure how to go about the two questions here and I have not been able to find something similar like this online. Would appreciate anyones help on how to answer the two questions! Thank you in advance.

Answer 1

When converting part a) to a fraction, you should get: $$\frac{67}{99}+\frac{94}{99}$$ If you divide by $99$, the digits on the numerator should repeat in the decimal form if the numerator is less than $100$ for positive integers.

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Let me demonstrate why: $$\frac{67}{99}=\frac{67}{100}+\frac{67}{9900}=0.67+\frac{67}{9900}$$ We can continue this process: $$\frac{67}{9900}=\frac{67}{10000}+\frac{67}{990000}=0.0067+\frac{67}{990000}$$ $$\frac{67}{990000}=\frac{67}{1000000}+\frac{67}{99000000}=0.000067+\frac{67}{99000000}$$ Repeating this process infinitely gives: $$\frac{67}{99}=0.67+0.0067+0.000067\cdots=0.\dot{6}\dot{7}$$

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So, try to convert the decimal forms you have into fractional form, and add the two fractions for part a) and multiply for part b).

If you wish, you can comment the answers you get below and I could verify them.