# Swimming pool water calcium level is 300 parts per million.

#### 4545

##### New member
The swimming pool water calcium level is 300 parts per million.

It needs to be 200 parts per million.

The replacement water has a calcium level of 100 parts per million.

What percent of the old water needs to be replaced?

Thanks.

#### mmm4444bot

##### Super Moderator
Staff member
The swimming pool water calcium level is 300 parts per million.

It needs to be 200 parts per million.

The replacement water has a calcium level of 100 parts per million.

What percent of the old water needs to be replaced?

There is no single formula that solves all word problems similar to this one. You are probably expected to (1) pick a symbol to represent what they're asking for, (2) consider this unknown quantity along with the given values and scenario until you realize relationship(s) between them, and then (3) use your symbol to write and solve an equation.

In case you're wondering, parts per million (ppm) is a ratio without units, just like percent is a ratio without units.

1% is 1/100th of the whole

1ppm is 1/1000000th of the whole

A nice thing about this exercise is that none of the volumes matter because everything is proportional. For example, any water removed is 300ppm -- whether you remove 1,000 liters or you remove one drop. The swimming pool could be olympic-sized (2.5 million liters) or backyard wading-size (100 liters) -- the answer will be the same. So focus on the parts in the original water and the parts in the replacement water, and resist the urge to think of volumes.

Here's a quasi-math description of what's happening.

Some percent of 300 parts is removed from the 300 parts, and then the same percent of 100 parts is added. The result is 200 parts.

Can you pick a symbol to represent this unknown percent and use it to write an equation that models what's happening?

Please show us what you tried, and we can go from there. :cool:

Also, kindly read the forum guidelines. Thank you!

#### 4545

##### New member
300 - 300x + 100x = 200

300 (- 300x + 100x) = 200

300 - 200x = 200

(300 - 300) - 200x = (200 - 300)

- 200x = - 100

(- 200x / -200) = (- 100 / - 200)

x = 0.5

x = 50%

([desired water ppm] - [old water ppm]) / ([water to be added ppm] - [old water ppm]) = decimal * 100 = percent to drain.

#### mmm4444bot

##### Super Moderator
Staff member
([desired water ppm] - [old water ppm]) / ([water to be added ppm] - [old water ppm]) = decimal * 100 = percent to drain [and replace].

Interesting. It's a simplified ratio of two different percent changes, where each percent change is from one ppm to another. And, because both percent change and ppm are ratios themselves, it's a ratio of ratios of ratios.

Looks like you have found a formula, for the particular mixture scenario taking place in this exercise.

Your algebraic solution also looks good.

#### 4545

##### New member
I think I found a better dilution formula, equation.

([desired water ppm] - [old water ppm]) / ([water to be added ppm] - [old water ppm]) = decimal * 100 = percent to drain.

200-300/100-300

-100/-200

([old water ppm] - [desired water ppm]) / ([old water ppm] - [water to be added ppm]) = decimal * 100 = percent to drain.

300-200/300-100

100/200

#### mmm4444bot

##### Super Moderator
Staff member
Both versions are basically the same.

If you wanted to increase the calcium concentration, you could still use either.

#### mmm4444bot

##### Super Moderator
Staff member
300 - 300x + 100x = 200

300 (- 300x + 100x) = 200

300 - 200x = 200

(300 - 300) - 200x = (200 - 300)

- 200x = - 100

(- 200x / -200) = (- 100 / - 200)
Hello, again. I had forgotten to mention a couple things about notation.

When you typed grouping symbols around the like-terms on the left-hand side (in your first step above), you placed the subtraction operator inside the grouping symbols (shown in red).

I realize that you were thinking in terms of negative 300x, but you need to show that you switched from subtraction to addition:

300 + (-300x + 100x)

Without this correction, your typing actually means that (-300x+100x) is being multiplied by 300. :cool:

Less important are some instances where you put space between a negative sign and its number.

I think -200 is better than - 200, as it helps to distinguish between when a hyphen represents a negative sign (i.e., a negation) versus when it represents a minus sign (i.e., a subtraction operator).

~ Cheers