| 1 |
percentages: parts and wholes |
|---|---|
| 2 |
percentages: measuring increments and decrements |
| 3 |
arithmetic with negative and positive numbers |
\( \frac{x}{y} \times 100 \)
this formula is used for finding out how much percentage \(x \) is out of \(y \). it's a mouthful, but it's simple.
example question: \(18 \) out of \(20 \) is how much? \(18 \) [the part] would be \(x \). \(20 \) [the whole] would be \(y \). then, just do \(\frac{18}{20} \times 100\), which gets you \(\frac{1800}{20} \), evaluating to \(90 \). so, \(18 \) is \(90 \)% of \(20 \).
\(z = x \times \left(1 \pm \frac{y}{100}\right) \)
this formula is used for finding the new value (\(z \)) after the old value (\(x \)) is increased/decreased by a certain percentage. (\(y \))
example question 1: if \(60 \) is decreased by \(40% \), what is it's new value? well, in this case, \(x = 60 \), since \(60 \) is the whole, and \(y = 40 \), as \(40 \) is the percentage/decrement. the equation would look like \(z = 60 \times \left(1 - \frac{40}{100}\right) \). \(\frac{40}{100} = 0.4 \), \(\ 1 - 0.4 = 0.6 \), and \(\ 60 \times 0.6 = 36 \), so \(\ z = 36 \), meaning that a \(40 \)% decrease in \(60 \) would change \(60 \) to \(36 \).
now, let's say this time \(60 \) is increased by \(40\)%. same math as last time, except the formula is \(z = 60 \times \left(1 + \frac{40}{100}\right) \). \(1 + 0.4 = 1.4\), and \(60 \times 1.4 = 84\), so a \(40\)% increase of \(60\) would amount to be \(84\).
\(z = \frac{x}{\left(1 \pm \tfrac{y}{100}\right)} \)
this formula is used for finding the old value (\(z\)) by using the new value (\(x\)) when giving a percentage increment/decrement (\(y\)). let's take a look.
suppose the value of \(z\) was decreased by \(30\)% to 50. in this case, you'd solve \(1 - \frac{30}{100}\). \(\frac{30}{100} = 0.3\) and \(1 - 0.3 = 0.7\). then, you'd have to solve \(\frac{50}{0.7}\). punch that into a calculator, and you get \(\frac{500}{7}\). so, \(z = \frac{500}{7}\).
now, let's say the value of \(z\) was increased by \(30\)%. this time, you'd have to solve \(1 + \frac{30}{100}\), which would get you \(1 \frac{30}{100}\), which is equivalent to \(1.3\). now, you have to solve \(\frac{50}{1.3}\), which gets you \(\frac{500}{13}\). so, \(z = \frac{500}{13}\).
\(z = \frac{y - x}{x} \times 100\)
this formula is used for measuring the percentage increase/decrease between 2 numbers, whereas \(z\) is the percentage change, \(y\) is the new value, and \(x\) is the old value. let's check it out.
suppose the old value (\(x\)) is \(85\), and the new value (\(y\)) is \(35\). now, what you could do is solve \(85 - 35 = 50\), but \(50\) is the difference between the two, not the percentage change. so, let's see what we'd actually do.
first, you'd substitue (replace) \(x\) with \(85\), and \(y\) with \(35\), so your new formula would be \(z = \frac{35 - 85}{85} \times 100\). well, \(35 - 85 = -50\), so you're left with \(z = \frac{-50}{85} \times 100\). simplify the fraction (use a calculator!!.. unless you're a genius like me.), and you get \(-\frac{10}{17}\). finally, you must solve \(-\frac{10}{17} \times 100\), which gets you \(-\frac{1000}{17}\). but wait! why is the fraction negative? well, the funny thing about this formula is; if the answer is negative, it represents a decrement, but if the answer is positive, it represents an increment. so, the percentage difference between \(85\) and \(35\) is a \(\frac{1000}{17}\)% decrease.
now, let's switch the values. the old value (\(x\)) is \(35\), and the new value (\(y\)) is \(85\). this time, the formula would be \(z = \frac{85 - 35}{35} \times 100\). \(85 - 35 = 50\), so now you're left with \(z = \frac{50}{35} \times 100\). simplify \(\frac{50}{35}\), and you get \(\frac{10}{7}\). \(\frac{10}{7} \times 100 = \frac{1000}{7}\). this time, the answer is positive, representing a percentage increment. so, the percentage increment between \(35\) and \(85\) is \(\frac{1000}{7}\)%.
multiplication and division