EnglishMath

Vocabulary
01

Vocabulary

Ratio

Ratio, in math, is a term that is used to
compare two or more numbers. It is
used to indicate how big or small a
quantity is when compared to another.
In a ratio, two quantities are compared
using division. Here the dividend is
called the 'antecedent' and the divisor
iscalled

the

'consequent'.

For

example, in a group of 30 people, 17
of them prefer to walk in the morning
and 13 of them prefer to cycle. To
represent this information as a ratio,
we write it as 17: 13.

Proportion

Proportion

isamathematical

comparison

between

two

numbers.

According to proportion,

if two sets of given numbers are
increasing

ordecreasing

inthe

same ratio, then the ratios are said
to be directly proportional to each
other.

Proportions

are

denoted

using the symbol

"::" or "=". For

example, the time taken by train to
cover 50km per hour is equal to the
time

taken

byittocover

the

distance of 250km for 5 hours. Such
as 50km/hr = 250km/5hrs.

Vocabulary

Algebraic
approach

Algebraic approach is a method to
solve a mathematical problem using
algebra.For example look at the
example six chapter three.

Cross-

multiplying

Cross-multiplying to clear an
equation of fractions when each
side consists of a fraction with a
single denominator by multiplying
the numerator of each side by the
denominator of the other side and
equating the two products obtained
Example :

Vocabulary

Simplifying

Simplifying In mathematics, simply
or simplification is reducing the
expression/fraction/problem

ina

simpler form. It makes the problem
easy

with

calculations

and

solving.
Example :

Vocabulary

Actual size

The true size of an object
represented

byascale

model or drawing.

Actual length

The true length of an object
represented

byascale

model or drawing.

Actual width

The true width of an object
represented

byascale

model or drawing.

Vocabulary

Direct variation

Direct variation and inverse variation are
two

types

of

proportionalities.

Proportionality refers to a relationship
where two quantities are multiplicatively
connected by a constant. In a direct
variation, the ratio of the two quantities
remains the same whereas in an inverse
variation the product of the two quantities
remains constant.

One-fourth

The

fraction

that

represents

the

real

number 0.25; ¼.

Video
02

Question

“Solve the proportion!

8
36=

10
𝑛”

Solution

Solution 1 : Equivalent ratios

Ratios both of numerator

equals to

Ratios both of denominator

8
36 = 10

𝑛

8 ×

𝟏𝟎

𝟖= 10

36 ×

𝟏𝟎

𝟖= 𝑛

80

8= 10

360

8= 𝑛

10 = 10

45 = 𝑛

Solution 2 : Equivalent ratios

Ratios both of numerator

denominator is the

same

8
36 = 10

𝑛

8 ×

𝟑𝟔

𝟖= 36

10 ×

𝟗

𝟐= 𝑛

8 ×

𝟗

𝟐= 36

90

2= 𝑛

72

2= 36

45 = 𝑛

Solution

Solution 3 : Cross-Multiplying

numerator
denominator= numerator

denominator

8
36 = 10

𝑛

8 × 𝑛 = 10 × 36

8𝑛 = 360

8𝑛
8=

360

8

𝑛 = 45

Solution 4 : Multiply both side of
the equation with n

8
36= 10

𝑛

8
36 × 𝑛 = 10

𝑛 × 𝑛

8
36𝑛 = 10

8
36 𝑛 × 36

8 = 10 × 36

8

𝑛 =360

8

𝑛 = 45

Example 8, 10, 13, and 14.

03

Example of Chapter 3

Example 8

On a blueprint for a new office
building,

the

rectangular

conference room measures 4.5
in by 12 in. The shorter wall of
the actual room measures 15 ft.
How

much

carpeting

will

be

needed to cover the floor of the
actual room?

Solution :
Assumption that :
The side of the room on the blueprint are proportional to
the actual sizes or the ratios of the corresponding sides
are equal.

The model

lenght on blueprint

actual length
=width on blueprint

actual width

4.5
15 = 12

𝑤

4.5 × 𝑤 = 12 × 15

4.5𝑤 = 180

4.5𝑤
4.5 = 180

4.5

𝑤 = 40
So, the carpeting will be
needed to cover the floor of
the actual room by 𝟔𝟎𝟎𝒇𝒕𝟐

Let w be the actual
width

The Area of the actual
room
= actual len𝑔𝑡ℎ
× actual width
= 15 × 40 = 600𝑓𝑡2

Example 10

A construction firm knows that the
number of new houses that it can build
in a month varies directly with the cost
of labor due to overtime charges. If 12
houses can be built when pays an
average of $14.50 per hour, how much
should the company expect to pay its
workers, if it need to build 20 houses?

Solution :
Since the quantities very directly, so the model
proportion is:

𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝒉𝒐𝒖𝒔𝒆𝒔

𝐂𝐨𝐬𝐭 𝐨𝐟 𝒍𝒂𝒃𝒐𝒓
= 𝟏𝟐

𝟏𝟒.𝟓𝟎

Let c be the cost of labor to build 20 houses.

20
𝑐=
12

14.50

12 × 𝑐 = 14.50 × 20

12𝑐 = 290

𝑐 =

290

12

𝑐 = $24.17 per hour

So, the company must pay $24.17 to
their workers, if it needs to build 20
houses.

Example 13

The floor of a social hall that is
45 ft by 82 ft is carpeted expect
for a circular dance floor in the
middle that has a diameter of 20
ft. To the nearest tenth of a
percent, what percent of the floor
is carpeted?

The area of the entire room
Area = length × width

= 45 × 82
= 3690𝑓𝑡2

The area of the circular dance floor
Area = 3.14 × circle radius2

= 3.14 × 102

= 3.14 × 100
= 314𝑓𝑡2

The carpeted area
= The area of the entire room
− The area of the circular dance floor
= 3690 − 314
= 3376 𝑓𝑡2

So, Jessica was twice as old as
Melissa 15 years ago.

Solution :
We know that
length of the entire room = 45 ft
width of the entire room = 82 ft

circle radius of the dance floor =20

2

= 10ft

The percentage model
Area of carpeted space

Area of entire room
=
𝑥

100

3376
3690 =
𝑥

100

3690 × 𝑥 = 3376 × 100

3690𝑥 = 337600

3690𝑥
3690 = 337600

3690

𝑥 = 91.5 %

Let x as the percent of the floor is
carpeted

So, the percent of the floor is carpeted is 91.5%

Example 14

A certain chemical solution used in
manufacturing batteries contains water
and acid. Intially, the solution is made
with 50 kg of dry acid and 900 L of
water. The solution is mixed and left to
sit so that the water evaporates until it
becomes a 30% acid solution. How
much water, the nearest liter, has to
evaporate for this to be accomplished?

Solution :
We know that
Total water = 900 L
Amount of acid = 50 kg

Percent of acid solution = 30%=30

100

Let x be the amount of water that
has evaporated

The amount of remaining water
= total water − 𝑥
= 900 − 𝑥

The total weight
= amount of remaining water + weight of dry acid
= 900 − 𝑥 + 50
= 950 − 𝑥

The model for the problem

Amount of acid
The total weight = 30

100

50

950 − x = 30

100

30 × 950 − 𝑥 = 50 × 100

28500 − 30𝑥 = 5000

28500 − 5000 = 30𝑥

30𝑥 = 23500

30𝑥
30 = 23500

30

𝑥 = 783.3

𝑥 = 783

So, 783 L of water has
to evaporate