In this article, we will discuss some basic arithmetic concepts. We will cover some mathematical concepts under Fractions, Decimals, Percents and Ratios. This article is not a cover-it-all or an introduction to these topics but rather it will address some concepts and properties of these topics.
We will discuss under this subtopic comparison of fractions when the numerators and/or denominators are changed.
Changing numerators, same denominators:
If a> b, then (a/c) > (b/c) (for c > 0) e.g
4/13 > 6/13, 3/50 > 0/50, 8/3 < 14/3
Changing denominators, same numerators:
If a > b, then c/a < c/b (for a > 0, b > 0, c > 0) .
Bigger denominators make fractions smaller e.g.
2/5 > 2/7, 7/24 > 7/36
Changing both denominators and numerators:
If the denominator gets smaller and the numerator gets bigger, then the fraction gets bigger. 3/8 < 4/7 for example.
What if both numerator and denominator are increased.
Note we can increase both by either multiplying with a positive integer or by adding a positive integer.
If we multiply both the numerator and denominator by the same positive integer, we get an equivalent fraction e.g 3/7 (multiply both by 2) = 6/14.
If we add the same positive integer to both the numerator and the denominator, then the resulting fraction is closer to 1 than was the original fraction on the number line. e.g
2/5, if we add 5 to both, we get 7/10. The resulting fraction 7/10 is greater than 2/5 because 2/5 is less than 5/5 or 1.
But what of 5/4, if we add 5 to both, we get 10/9 which is less than 5/4 because 5/4 is greater than 5/5 or 1.
Note in both preceding examples, we added the same number to both numerator and denominator i.e 5 added up, 5 added below, this is where the 1 comes from since 5/5 is 1. And so in both cases, the resultant fractions get closer to 1 on the number line.
What if we add different numbers to both the numerator and the denominator? What effect does this have on the resultant fraction?
For instance, let’s add 2 to the numerator and 5 to the denominator. The resulting fraction is closer to 2/5 on the number line e.g if we start with 1/8, we get (1+2)/(8+5) = 3/13.
3/13 > 1/8 because 1/8 is less than 2/5 on the number line
However, if we start with 3/4, we get (3+2)/(4+5) = 5/9.
5/9 < 3/4 because 3/4 is greater than 2/5 on the number line.
There is a pattern here which we can summarize as follow:
If we start with a fraction a/b and then add c to the numerator and d to the denominator, the resultant fraction (a+c)/(b+d) will be closer to c/d than was the original fraction a/b.
If a/b < c/d, then a/b < (a+c)/(b+d) .
If a/b > c/d, then a/b > (a+c)/(b+d).
This is the central attractor effect where c/d is the central attractor.
What happens when we subtract rather than add the numbers from both the numerator and denominator?
The effect is the opposite which is the central repeller effect where c/d is the central repeller.
If a/b < c/d, then a/b > (a-c)/(b-d) .
If a/b > c/d, then a/b < (a-c)/(b-d).
And finally, if we add to the numerator and subtract from the denominator or vice versa, then whichever fraction has the bigger numerator and the smaller denominator is bigger.
Working with Percents
Approximate conversion from fraction to percents
Sometimes we might need to quickly approximate percents from fractions in our head, well we can use number sense.
What we really need to do is find a way to move the denominator to or closer to 100. The principle here is if we divide or multiply the numerator and denominator by the same number, the resultant fraction is the same.
Since percents are simply from fractions with denominators of 100. We can use roughly approximate a percent from fraction if we move the denominator to or closer to 100. e.g.
8/33, looking at the denominator 33, multiplying by 3 gives 99 which is approximately 100 so the fraction can be converted to 24/99 > 24/100 = 24%. So 8/33 is roughly 24%.
11/44 = 77/98 > 77/100 = 77%, therefore 11/44 is roughly 77%.
The decimal form of a percent is called the multiplier for that percent. This is so because we can multiply a number by this decimal form to get the percent of the number. E.g decimal form of 80% is 0.8 which is the percent multiplier of 80%.
What is 80% of 200 -> 0.8 * 200 = 160.
240 is 30% of what number? -> 240 = 0.3 * x -> x = 240/0.3 = 800.
Percent Increases and Decreases
Percent Increase could be phrased in many ways e.g A increased by x% or B is x% greater than A. Either way, this simply means a percent increase for A.
If A increases by x%, the whole original part is still there plus an extra x%. Therefore the multiplier for an x% increase is 1+ x/100. Note x/100 is the decimal form of x%.
In general, if a statement talks about an x% increase, the multiplier is as follow:
(multiplier for an x% increase) = 1 + (x% as a decimal).
If an item originally cost $800 and the price increased by 20%, the new price can be calculated by using the multiplier. The multiplier for a 20% increase is 1.2 (1+0.2). Thus new price is 800*1.2 = $960.
After a 30% increase, the price of something is $78. What was the original price?
The multiplier for a 30% increase is 1.3 (1+0.3), so Original Price(OP) * 1.3 = 78.
OP = 78/1.3 = 780/13 = $60.
Percent Decrease could be phrased in many ways e.g A decreased by x% or A (price of something) is discounted by x% or B is x% less than A. Either way, this simply means a percent decrease for A.
If A decreases by x%, the whole original part is still there minus an extra x%. Therefore the multiplier for an x% decrease is 1- x/100 (x/100 is the decimal form of x%).
In general, if a statement talks about an x% decrease, the multiplier is as follow:
(multiplier for an x% decrease) = 1 – (x% as a decimal).
Thus, for example, the multiplier for a 28% decrease is 1-0.28 = 0.72.
(1) A $1700 item is discounted 30%. What is the new price?
multiplier = 1 – 0.3 = 0.7
New Price = 170 * 0.7 = $119.
(2) After an item was discounted 80%, the new price is $150. What was the original price?
multiplier = 1 – 0.8 = 0.2
OP * 0.2 = 150 -> OP = 150/0.2 = 1500/2 = $750
Finding the percent
In some cases, we might have the starting and ending values and need to find the percent of the increase or decrease.
Since new value(new) = multiplier(m) * old value(old), m = new/old.
If we know the multiplier, we can easily find the percent increase or decrease. It will be an increase if the multiplier is greater than 1 or a decrease if it’s less than 1.
(1) The price of an item increase form $60 to $102. What was the percent increase?
m = new/old = 102/60 = 17/10 = 1.7 = 1+0.7 = 1+70%.
Thus the percent increase is 70%.
(2) The price of an item decrease from $250 to $200. What was the percent decrease?
m = new/old = 200/250 = 20/25 = 80/100 = 0.8 = 1 – 0.2 = 1 -20%.
Thus the percent decrease is 20%.
(3) The price of an item increase from $200 to $800. What was the percent increase?
m = new/old = 800/200 = 4 = 1+3 = 1+ (300/100) = 1 + 300%.
Thus the percent increase is 300%.
Sequential Percent Changes
Sometimes a value might increase or/and decrease by multiple percents in sequential form. e.g A might increase by x% and then decreased by y% sequentially, what will be the final value? We will simply multiply the initial value by all the multipliers in succession e.g
(1) An item initially cost $100. At the beginning of the year, the price increased by 30%. After the increase, an employee purchased it with a 30% discount. What price did the employee pay?
m for a 30% increase is 1.3, m for a 30% decrease is 0.7.
Price paid = 100*1.3*0.7 = $91.
(2) At the beginning of the year, the price of an item increased by 30%. After the increase, an employee purchased it with a 40% discount. The price the employee paid was what percent below the original price?
m for a 30% increase is 1.3, m for 40% decrease is 0.6.
Total of multipliers is 1.3*0.6 = 0.78 = 1 – 0.22(22%).
That’s a total 22% decrease below the original price.
(3) The price of the stock increased by 20% in January, dropped by 50% in February and increased by 40% in March. Find the percent change for the 3-month period.
Using all the multipliers, 1.2*0.5*1.4 = 0.84 = 1 – 0.16
Percent change is 16% decrease.
Working with Ratios
Percents and Ratios
Percents and ratios tell us the same type of information. Both of them tell us about fractions in a whole but without the actual information or numbers, they tell us nothing about absolute values.
For example, the 2 statements below are the same:
- 20% of employees are programmers, while the remaining 80% are customer service personnel(CSP).
- The ratio of CSP to programmers is 4:1.
But neither allows us to know the actual numbers of employees or programmers or CSPs.
Ratios are represented in the following forms
- a to b form: e.g ratio of male to female is 5 to 6.
- fraction form: e.g ratio of male to female is 5/6.
- colon form: e.g ratio of male to female is 5:6.
- idiom form: e.g For every 5 males, there are 6 females.
Sometimes we have to relate ratios between different subgroups in a collection together. We might be given separate ratios and we will have to relate them together to the whole or to absolute quantities.
We can combine ratios when we are given 2 or more separate ratios within a large group.
One way to approach this is to find equivalent fractions for each ratio so that the common term comes to be represented by the same number in both ratios.
Let’s see this in some examples:
(1) A certain high school team has only sophomores, juniors and seniors. The ratio of sophomores to juniors is 2:3 and ratio of juniors to seniors is 5:6. Sophomores are what fraction of the whole team?
Notice here that
Sophomores(So) to Juniors(Jr) ratio is 2:3 -> So:Jr = 2:3
Juniors(Jr) to Seniors(Sr) ratio is 5:6 -> Jr:Sr = 5:6
Notice both ratios both have Jr in common, so we can find equivalent fractions that will make both ratios have common Jr terms.
So:Jr = 2:3 = 10:15
Jr:Sr = 5:6 = 15:18
So the combined ratio So:Jr:Sr = 10:15:18
Whole = 10+15+18 = 43
So:Whole = 10/43
(2) In a certain company, the ratio of programmer to marketers is 3:8, and the ratio of customer sevice reps(CSRs) to marketers is 2:3. If there are 27 programmers, there are how many CSRs?
P:M = 3:8, C:M = 2:3, M is common so we find equivalent terms (essentially finding the LCM of 8 and 3 which is 24)
P:M = 3:8 = 9:24
C:M = 2:3 = 16:24
combined P:C:M = 9:16:24.
Now it’s obvious ratio P:C = 9:16, if there are 27 P(programmers)
27/C = 9/16 -> C = 27*16/9 -> C = 48
CSRs = 48
(3) 1 cup of butter is enough for 8 of Smith’s cookies and 1 cup of sugar is enough for 5 of these cookies. If he used 15 more cups of sugar than butter, how many cookies did he make?
We have cookies(C), butter(B) and sugar(S) in this problem.
B:C = 1:8 = 5:40
S:C = 1:5 = 8:40
B:S:C = 5:8:40
B:S = 5:8
B = S + 15 (15 more cups of sugar than butter), so now we can find S.
(S+15)/S = 5/8 -> 8(S+15) = 5S -> 8S + 8*15 = 5S -> 3S = 8*15
S = 8*15/3 = 40.
From the S:C ratio above S/C = 1/5
40/C = 1/5
C = 40*5 = 200
He made 200 cookies.
In this article, we have discussed Fractions, Decimals, Percents and Ratios and few of their properties.
In the next math module, we will discuss Integer properties.