The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then $$\frac{y}{x}$$ is equal to:

We have ten numbers already arranged in increasing order:

$$10,\;22,\;26,\;29,\;34,\;x,\;42,\;67,\;70,\;y$$

The question tells us that the mean (arithmetic average) of these ten numbers is $$42$$. The definition of the mean for $$n$$ numbers is

$$\text{Mean} \;=\;\dfrac{\text{Sum of all } n \text{ numbers}}{n}.$$

Here $$n=10$$, so

$$\dfrac{\text{Sum of the ten numbers}}{10}=42.$$

Multiplying both sides by $$10$$ gives the total sum:

$$\text{Sum of the ten numbers}=42 \times 10=420.$$

Next we add up the eight known numbers:

$$10+22+26+29+34+42+67+70$$

First pair: $$10+22=32$$; now $$32+26=58$$; add $$29$$ to get $$87$$; add $$34$$ to reach $$121$$; add $$42$$ to obtain $$163$$; add $$67$$ to reach $$230$$; finally add $$70$$ to get $$300$$.

Thus the sum of the known numbers is $$300$$. Letting the unknowns stand,

$$300 + x + y = 420.$$

Subtracting $$300$$ from both sides gives

$$x + y = 420 - 300 = 120.$$

Now we use the information about the median. For an even number of observations (here $$10$$), the median is the average of the 5th and 6th terms in the ordered list. The 5th term is $$34$$ and the 6th term is $$x$$, so the median is

$$\dfrac{34 + x}{2}.$$

We are told that this median equals $$35$$. Therefore,

$$\dfrac{34 + x}{2} = 35.$$

Multiplying by $$2$$ on both sides,

$$34 + x = 70.$$

Subtracting $$34$$ gives

$$x = 70 - 34 = 36.$$

With $$x$$ known, substitute in the earlier relation $$x + y = 120$$:

$$36 + y = 120.$$

Subtracting $$36$$ from both sides,

$$y = 120 - 36 = 84.$$

Finally, we compute the required ratio $$\dfrac{y}{x}$$:

$$\dfrac{y}{x} = \dfrac{84}{36}.$$

Both numerator and denominator are divisible by $$12$$, so we simplify:

$$\dfrac{84 \div 12}{36 \div 12} = \dfrac{7}{3}.$$

Hence, the correct answer is Option B.