Chapter 3- Learning Outcome 3_Mathematics for Technologists

21st
Century Mathematics for
the Industrial Technology
Learners: Automotive
Technology
BABY 'RLENE I. BAYOC

Fractions,
Ratio and Proportion
Chapter
III
21st
Century Mathematics for the Industrial Technology Learners: Automotive Technology

CHAPTER III: Fractions, Ratio and Proportion
Learning
Outcomes:
At the end of
the chapter,
the student
must have:
LO1. Defined, compared, and simplified
common fractions and perform arithmetic and
conversion with fractions;
LO2. discussed and solved problems involving
ratio and proportion; and
LO3. analyzed problems to determine
whether they are direct or inverse proportions
to set up proportions, and solve for unknowns.

Proper fraction
Division
Dividend
Improper fraction
Rational
number
FRACTIONS
Denominator
Numerator
n
u
m
b
e
r Edge
Dodecahedron
Divisor
Inverse
Proportions/Variation
s or Indirect
Proportions
Fraction.
Direct
proportion/variations

Thank you

Learning
Outcome 1:
Defined, compared, and simplified
common fractions and perform
arithmetic and conversion with
fractions;

A fraction is defined as part of an entire object. Formally defined,
if p and q are algebraic expressions, the quotient, or ratio, p/q is
called a fractional expression or a fraction with numerator p and
denominator q, read as p over q. Note that the denominator of a
fraction cannot be zero. If q=0, the expression p/q is not defined.
Therefore, whenever we use a factional expression, we shall
automatically assume that the denominator is nonzero.
Fractions
Define, compare, and
simplify common
fractions

Fractions
A fraction is proper when the numerator is less than the
denominator or improper when the numerator is equal or greater than
the denominator also similar (like) fractions if they have the same
denominator or dissimilar (unlike) fractions when they different
denominator. Two fractions are equivalent fraction if they have the
same value. To compare similar fractions, the fraction with the
greatest numerator is the largest fraction and the fraction with the
least numerator is the smallest fraction. When comparing dissimilar
fractions, we have different solutions. First is by division; second is by
making them similar fractions by the LCD (least common
denominator); and third is by cross multiplication.
Fractions
Define, compare, and
simplify common
fractions

Solutions in Comparing Dissimilar Fractions.

There are rules that we have to follow when applying the four fundamental
operations with fractions, likewise in conversion of fractions to decimals and
percent.
1. Adding/Subtracting Fractions with the Same Denominator:
To add/subtract two fractions with the same denominator, add/subtract the
numerators for the numeratorand copy the denominator.
Example 1. 3/11 + 2/11
= (3+2)/11
= 5/11
Example 2. 3/11 – 2/11
= (3 -
2)/11
= 1/11

2. Adding/subtracting Fractions with Different Denominators:
To Add/subtract Fractions with different denominators, find the Least
Common Denominator (LCD) of the fractions. Rename the fractions to
have the LCD. Add/subtract the numerators of the fractions. Simplify
the fraction until it is in its simplest form.
Example 1. 3/5 + 2/15
= (9+2)/15
= 11/15
Example 2. 3/8 - 2/12
= (9 -
4)/24
= 5/24

3. Multiplying fractions: To multiply fractions, multiply the
numerators then multiply the denominators. Then simplify if
possible.
Example 1. i/r x f/z
= if/rz
Example 2. 2/8 x 5/12
= 10/96
= 5/48

4. Dividing fractions: To divide fractions, multiply the extremes over the means or
use the reciprocal of the divisor then proceed to multiplication. Simplify your final
answer.
Note that before multiplying or dividing fractional expressions in general, try to factor
the numerators and denominators completely to reveal any factors that can be
canceled.
Fractions
Define, compare, and simplify common
fractions
Example 1. i/r ÷ f/z
= iz/rf
Example 2. i/r ÷ f/z
= i/r x z/f
= iz/rf

5. Converting Fractions to Decimals:
A. The simplest method is to use a calculator by just dividing the numerator by the
denominator.
B. The method used when there is no calculator is by long division.
C. Another Method is to make the denominator equal to 10, 100, 1000 ...
Step 1: Find a number that when you multiply your denominator by that number then
your denominator becomes 10, 100, 1000, .... or any 1 followed by 0’s.
Step 2: Multiply both the numerator and the denominator by that number.
Step 3. Then count the number of zero/es in your denominator and it would be
the number of decimal places in your numerator.
Example 1. 3/5 = 3/5 x 2/2 = 6/10 = 0.6
Example 2. 50/250 = 50/250 x 4/4 = 200/1000 = 0.2

5. Converting Fractions to Decimals:
The diagram represents the
conversion of fractions to
decimals, decimals to percent,
percent to fractions and percent to
decimals.
Note that if we convert
fractions to percent, we have to
convert it first to decimals.
For decimals to fractions,
convert it first to percent then to
fraction. There is other way to do
this, write it the way you read it.

6. Converting recurring decimals to fractions:
Step 1: Let x = recurring decimal in expanded form.
Step 2: Let the number of recurring digits = n.
Step 3: Multiply recurring decimal by 10n.
Step 4: Subtract (1) from (3) to eliminate the recurring part.
Step 5: Solve for x, expressing your answer as a fraction in its simplest form.
Examples of recurring decimal are 1/3 = 0.333333..., 1/7 = 0.142857142857...
A recurring decimal exists when decimal numbers repeat forever.

6. Converting recurring decimals to fractions:
Example 1. Convert 0.111111…
to fraction.
Solution: Let x = 0.1111111… = 0.1̅
10 x = 1.1̅
10x – x = 1.1̅ - 0.1̅
9x = 1
X = 1/9
Example 2. Convert 0.1212121212…
to fraction.
Solution: Let x = 0.1212121212… = 0.1̅2̅
100x = 12.1̅2̅
100x – x = 12.1̅2̅ - 0.1̅2̅
99x = 12
x = 12/99 or 4/33.

REFERENCES
https://www.emathzone.com/tutorials/algebra/multiplication-and-division-of-
fractions.html#ixzz6K8M3Cn7b
https://www.emathzone.com/tutorials/algebra/addition-and-subtraction-of-
fractions.html#ixzz6K8LmeWJl
https://www.cimt.org.uk/projects/mepres/book7/bk7i17/bk7_17i2.htm
https://www.mathsisfun.com/converting-fractions-decimals.html
https://www.aaamath.com/fra16_x2.htm
https://www.emathzone.com/tutorials/algebra/fractions.html#ixzz6K8LMdp5x
https://www.khanacademy.org/math/arithmetic/fraction-arithmetic
https://www.onlinemathlearning.com/recurring-decimals-fractions.html

Learning
Outcome 2:
discussed and solved problems
involving ratio and proportion; and

Ratio is the quantitative relation between two amounts showing the number of
times one value contains or is contained within the other. Ratio of two values a
and b is written as a : b or a/b or a to b.
For instance, the ratio of female BAT students to male BAT student is 3:25.
Here, 3 and 25 are not taken as the exact count of the BAT students but a
multiple of them, which means the female students can be 3 or 6 or 9... and the
male students is 25 or 50 or 75... It also means that in every twenty-eight BAT
students, there are three female students and twenty-five male students.
Rati
o

A proportion is a comparison of two ratios. If a : b = c : d, then a, b, c, d are
said to be in proportion and written as a : b :: c : d or a/b = c/d. Where a, d
are called the extremes and b, c are called the means. For a proportion
a : b = c : d, product of means = product of extremes → b*c = a*d.
Proportion can be used to solve problems on ratio.
Proportio
n
Proportion
Define, compare, and simplify common
fractions

Let us take a look of this example:
There is 40 liters of a solution (water-based paint) which has 40% paint. Extra paint
is added to it to make it to 60% paint solution. How much water has to be added
further to bring it back to 40% paint solution?
Solution: This question has 3 parts.
In the first part, there is 40% of paint in 40 L of solution → 16 L of paint in 24 L of
water = 40 L of solution. Let’s note the details in a table for better clarity and
understanding.
Paint Water Total
% Quantity in
L
% Quantity in L
40% 16 L 60% 24 L 40 L

In the second part, Extra paint is added to bring it to 60% of paint solution. Let
the amount of paint added be x liters.
Paint Water Total
% Quantity in L % Quantity in L
40% 16 L 60% 24 L 40 L
60% 16 L + x 40% 24 L 40 L + x

Then, (16 L + x)/(40 L + x) = 60/100 → x = 20 L of paint is added. Now, there is
16+20 = 36 L of paint in 24 L of water in a total of 60 L of solution. Third step is to
bring the paint percentage back to 40%, extra water is added and the amount of
paint remains the same. Let this extra amount of water be y liters.
16 L of paint constitutes 20% of the solution → 36/(60 L + y) = 40/100 → y = 30.
Therefore, 30 liters of water has to be added to the solution if paint has to be 40%
of paint for the whole solution of 90 L.
Paint Water Total
% Quantity in L % Quantity in L
40% 16 L 60% 24 L 40 L
60% 36 L 40% 24 L 60 L
40% 36 L 60% 24 L + y 60 L + y

REFERENCE
https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-
write-and-solve-proportions/e/constructing-proportions-to-solve-application-problems
https://www.onlinemathlearning.com/ratio-problems.html
https://www.mbacrystalball.com/blog/2015/08/28/ratio-proportion-problems/
https://brilliant.org/wiki/ratio-and-proportion/

Learning
Outcome 3:
analyzed problems to determine
whether they are direct or inverse
proportions to set up proportions,
and solve for unknowns.

If there is a constant k such that y = kx, holds for all values of x, we say that y is
directly proportional to x or that y varies directly as x (or with x). The constant k is
called the constant of proportionality or the constant of variation.
Example :
1. F is directly proportional to R, when F is equal to 5 and R is equal to 2. Find the
value of F when R is equal to 12.
Solution: Write the direct proportion → F R → F = kR
∝
Since F = 5 and R = 2 then 5 = k2 → k = 5/2. So, our equation is F = (5/2)R.
When R = 12, we have F = (5/2)(12) =30.
Direct
Proportion/Variation

Example 2.
A is directly proportional to the cube of B, when A = 81 and B = 3. Given B =5,
what is the value of A?
Solution: Write the direct proportion → A B3→ A = kB3
∝
Since A = 81 and B = 3 then 81 = k3
3 → k = 81/27 = 3. So, our equation is A = 3B.
When B = 5, we have A = 3(53) = 3(125) = 375.

If there is a constant k such that y=kx, for all nonzero values of x, we say that y is
inversely proportional to x or that y varies inversely as x (or with x).
Note that if x is inversely proportional to y then it is equal to x is directly proportional
to 1/y. We can rewrite it of these forms x 1/y → x = k (1/y)
∝
Example 3. A is inversely proportional to B. When A is 10, B is 2. Find the value of A
when B is 8.
Solution: Write the inverse proportion → A 1/B → A = k(1/B)
∝
Since A = 10 and B = 2 then 10 = k(1/2) → k = 20. So, our equation is A = 20 (1/B).
When B = 12, we have A = (20)(1/12) = 5/3.
Inverse Proportion/Variation or Indirect
Proportion

Example 4. A is inversely proportional to twice a number B. When A is 10, B is 2.
Find the value of A when B is 8.
Solution: Write the inverse proportion → A 1/2B → A = k(1/2B)
∝
Since A = 10 and B = 2 then 10 = k(1/4) → k = 40. So, our equation is A = 40
(1/2B).
When B = 12, we have A = (40)(1/24) = 5/3.

REFERENCE
https://www.emathzone.com/tutorials/algebra/mathematical-models-and-idea-of-direct-
and-inverse-variation.html#ixzz6K8HRS081
https://www.emathzone.com/tutorials/algebra/mathematical-models-and-ideaof-direct-
and-inverse-variation.html#ixzz6K8HH14xm
https://www.onlinemathlearning.com/proportions.html
https://www.splashlearn.com/math-vocabulary/division/numerator

Chapter 3- Learning Outcome 3_Mathematics for Technologists

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