Improve Your Algebra Skills with Practice Problems

Proof of an $\frac{a}{b}=\frac{c}{d}⟺\frac{a+b}{a-b}=\frac{c+d}{c-d}$
but going from $\frac{a}{b}=\frac{c}{d}$
I already proved it backwards
$\frac{a+b}{a-b}=\frac{c+d}{c-d}$ I multiply each side by (a-b)(c-d)
$ac-<!--\; -\; -->ad+bc-<!--\; -\; -->bd=ac-<!--\; -\; -->bc+ad-<!--\; -\; -->bd$ so $2bc=2ad$ which by dividing both sides by $2bd$ gives us :
$\frac{a}{b}=\frac{c}{d}$
any ideas how to proceed ?
Edit : this equivalence only works when $a\ne <!--\; \ne \; -->b$ and $c\ne <!--\; \ne \; -->d$

Some doctors recommend that men drink 3 liters of water every day . There are approximately 29.6 milliliters in 1 fluid ounce. Which measurement is closest to the number of fluid ounces in 3 liters ?
A. $89\text{ fl oz}$
B. $101\text{ fl oz}$
C. $10\text{ fl oz}$
D. $33\text{ fl oz}$

What is 2.8 as a fraction?

Missy used a coupon 20% off the price of her meal at a buffet restoraunt and received $3 discount. What was the original price of Missy's mea; before the discount?

How many inches are equal to 10 centimeters? Remember, 2.54 cm = 1 in. Round your answer to the nearest hundredth. Do not include units in your answer.

A geometric inequality in a triangle
If $x,y,z>0,$ could it be:
$\frac{z(2z+x)}{z+x}{b}^2+\frac{y(2y+x)}{y+x}{c}^2>\frac{yz}{y+z}{a}^2$
for a triangle $ABC,$ with $a=BC,$$b=AC,$$c=AB?$
If $b^2+c^2=a^2$, then the inequality holds, since
$\frac{z(2z+x)}{z+x}-<!--\; -\; -->\frac{yz}{y+z}=\frac{z^2(x+y+2z)}{(z+x)(y+z)},\frac{y(2y+x)}{y+x}-<!--\; -\; -->\frac{yz}{y+z}=\frac{y^2(x+2y+z)}{(z+x)(y+z)}$
If $b^2+c^2>a^2$, then the inequality holds, since
$\frac{z(2z+x)}{z+x}{b}^2+\frac{y(2y+x)}{y+x}{c}^2>\frac{yz}{y+z}{b}^2+\frac{yz}{y+z}{c}^2>\frac{yz}{y+z}{a}^2$
I can’t handle the case $b^2+c^2<a^2$ and I need some help

Tealeaf, a company that offers management solution to companies that sell online, announced in a 2008 consumer survey that nearly 9 out of 10 customers reported problems with transactions online. The survey sampled 2010 adults in the United States, 18 years and older, who had conducted an online transaction in the past year. Estimate the number of adults who reported problems with transactions online.

From 1992 to 1995, a certain country experienced a yearly inflation rate of 255%. Determine the price of 3.62 fast-food meal after 8 years at a 255% inflation rate.

1.Determine the number of pounds of Julie’s body weight that are fat. Julie is your client that is 59kg and 22% body fat-Determine your lean body mass (FFM) in lbs-Subtract 1.0 from Julie’s desired percent body fat of 18%._____% (desired percent body fat) 1.0 - ._____ = ._____
-Determine your recommended body weight for Julie at that new % body fat.
2.How many pounds would she have to lose or gain to get to your recommended body weight?
2b.Do you think this is realistic, healthy goal? Why or why not?
3.What changes can Julie make in her diet and exercise routine to improve her body composition (to maintain or get to your recommended body weight)?

Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups. Write an expression to represent the time it takes Edson to exercise as the sum of three different terms. Then simplify the expression.

partial fraction decomposition of $\frac{(2l+1)(l+1)}{K+(l+1)(l+2)(l-<!--\; -\; -->1)}$
I am struggling with the partial fraction decomposition of
$\frac{(2l+1)(l+1)}{K+(l+1)(l+2)(l-<!--\; -\; -->1)},$
where $K$ is a positive real number.
I would like to decompose it in a sum of simpler polynomial fractions.
I tried the Mathematica's function "Apart" but it is returning the fraction so I am thinking there is no simple partial fraction decomposition.
Could you help me?

Prove that $\frac{1}{15}<\frac{1}{2}\ast <!--\; \ast \; -->\frac{3}{4}\ast <!--\; \ast \; -->\cdots <!--\; \cdots \; -->\ast <!--\; \ast \; -->\frac{99}{100}<\frac{1}{10}$
My attempt:If we name the value $A$ we have:
$A^2<(\frac{1}{2}\ast <!--\; \ast \; -->\frac{3}{4}\ast <!--\; \ast \; -->\cdots <!--\; \cdots \; -->\ast <!--\; \ast \; -->\frac{99}{100})(\frac{2}{3}\ast <!--\; \ast \; -->\frac{4}{5}\ast <!--\; \ast \; -->\dots <!--\; \dots \; -->\frac{100}{101})=\frac{1}{101}\Rightarrow <!--\; \Rightarrow \; -->A<\frac{1}{10}$
But I don't know, how to prove the other side?

Give your answer accurate to 3 decimal places. Water is flowing at the rate of 9 ft3/min into a tank that is in the shape of a right circular cylinder whose base radius is 3 ft. How fast (in feet per minute) is the water level rising?

Which fraction is equivalent to $-<!--\; -\; -->\frac{4}{5}$? (A) $-<!--\; -\; -->\frac{-<!--\; -\; -->4}{-<!--\; -\; -->5}$
(B) $-<!--\; -\; -->\frac{4}{-<!--\; -\; -->5}$
(C) $-<!--\; -\; -->\frac{-<!--\; -\; -->4}{5}$

If a regular-size drinking glass holds about 0.25 liter of liquid, how many glasses can be filled from a 750-milliliter container?

Solve the exponential equation.
$7\cdot <!--\; \cdot \; -->5^{3x-<!--\; -\; -->1}+3=15$

What formula or rule has been used here?
I was in between proving a trigonometric identity but couldn't succeed. I went through the solution and saw this in between
$\begin{array}{rl}\frac{\cos <!--\; \; -->A\cos <!--\; \; -->B}{\sin <!--\; \; -->A\sin <!--\; \; -->B}& =\frac{3}{1}\\ \\ \frac{\cos <!--\; \; -->A\cos <!--\; \; -->B+\sin <!--\; \; -->A\sin <!--\; \; -->B}{\cos <!--\; \; -->A\cos <!--\; \; -->B-<!--\; -\; -->\sin <!--\; \; -->A\sin <!--\; \; -->B}& =\frac{3+1}{3-<!--\; -\; -->1}\end{array}$
What happened there in the second step?

Solve by factoring
$(s+9{)}^2-<!--\; -\; -->9=0$

Translate the following phrase into an algebraic expression. Do not simplify. Use the variable names "x" or "y" to describe the unknowns.

Fractional/rational form of $\mathrm{0.999...}$
Is it possible to express $\mathrm{0.999...}$, a repeating number, as a fraction? Or as a ratio of two numbers?
Basically all (my) attempts at the problem cancels all the terms and returns $1$. Is it even possible? Or has it been proven to be an exercise of futility?