How Do the Fractions A and B Differ in Value and Structure?

[Albert1]

A=$\dfrac {\sqrt {998}+9}{\sqrt{998}+8}$

B=$\dfrac {\sqrt {999}+9}{\sqrt{999}+8}$

Please compare A and B

 

[anemone]

A=$\dfrac {\sqrt {998}+9}{\sqrt{998}+8}$

B=$\dfrac {\sqrt {999}+9}{\sqrt{999}+8}$

My solution:

Spoiler

If we let $f(x)=\dfrac {\sqrt {x}+9}{\sqrt{x}+8}$, differentiate it w.r.t $x$ we get $f'(x)=\dfrac {-1}{2\sqrt{x}(\sqrt{x}+8)^2}$, i.e. $f'(x)<0$ for all real $x$, or more specifically, $f'(x+1)<f'(x)$ and this implies $f(x)>f(x+1)$, hence, we can say that $A=f(998)=\dfrac {\sqrt {998}+9}{\sqrt{998}+8}>B=f(998+1)=\dfrac {\sqrt {999}+9}{\sqrt{999}+8}$.

 

[Albert1]

A=$\dfrac {\sqrt {998}+9}{\sqrt{998}+8}$

B=$\dfrac {\sqrt {999}+9}{\sqrt{999}+8}$

Please compare A and B

$ let \,\,x=998$

$A-B$ = $\dfrac{1}{(\sqrt {x}+8)(\sqrt{x+1}+8\big)}\times \big [(\sqrt{x}+9)(\sqrt{x+1}+8\big)- (\sqrt{x+1}+9)(\sqrt{x}+8\big )\big ]$

=$\dfrac{1}{(\sqrt {x}+8\big)(\sqrt{x+1}+8 \big)}\times (\sqrt {x+1} -\sqrt {x}\big)>0$

$\therefore A>B$

 

[soroban]

Hello, Albert!

$A\:=\:\dfrac{\sqrt {998}+9}{\sqrt{998}+8}$

$B\:=\:\dfrac{\sqrt {999}+9}{\sqrt{999}+8}$

$$\text{Compare }A\text{ and }B.$$

Let $$x = 998$$

. . . . . . . . . . . . . . . . $$A \:\gtrless\:B$$

. . . . . . . . . . . $$\frac{\sqrt{x}+9}{\sqrt{x}+8} \;\gtrless\;\frac{\sqrt{x+1}+9}{\sqrt{x+1}+8}$$

$$\left(\sqrt{x}+9\right)\left(\sqrt{x+1}+8\right) \;\gtrless\;(\sqrt{x}+8)(\sqrt{x+1}+9)$$

$$\sqrt{x(x+1)} + 8\sqrt{x} + 9\sqrt{x+1} + 72$$
. . . . . . . . . . . . $$\gtrless\:\sqrt{x(x+1)} + 9\sqrt{x} + 8\sqrt{x+1} + 72$$

. . . . . $$8\sqrt{x} + 9\sqrt{x+1} \:\gtrless\:9\sqrt{x} + 8\sqrt{x+1}$$

. . . . . . . . . . . .$$\sqrt{x+1} \;{\color{red}>}\; \sqrt{x}$$

$$\text{Therefore: }\:A \:>\:B$$

 

[CellCoree]

:

A and B are both fractions with similar structures, but they have different numbers inside the square root symbols. A has 998 inside the square root while B has 999. This may seem like a small difference, but it can actually have a significant impact on the overall value of the fractions. By taking the square root of a larger number, B has a slightly higher value than A. This can be seen by simplifying both fractions to their decimal forms. A simplifies to approximately 1.005 while B simplifies to approximately 1.006. Thus, B has a slightly higher value than A. However, in terms of their overall structure and operations, A and B are essentially the same. Both fractions have a numerator that is the sum of a square root and a constant (9), and a denominator that is the sum of a square root and a different constant (8). Therefore, although B may have a slightly higher value, the overall comparison of A and B is that they are very similar in structure and operations, but differ in the specific numbers used.