Mathematics Therapy

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From: Samuel Omoogah
Date: Thu, Sep 16, 2021, 11:03
Subject: LINEAR EQUATIONS AND INEQUALITIES
To:

Example 1:

Solve 1 3

------ - -------- =0

1-2x 24+x

Solution:

The common denominator of the two fractions is (1-2x)(24+x). Multiply each term by (1-2x)(24+x)

1×(1-2x)(24+x)-3×(1-2x)(24+x)

----------------------- ----------------------- =

(1-2x) (24+x)

0×(1-2x)(24+x

Divide by using(24+x) in the denominator to divide(24+x) of the numerator and (1-2x) of the denominator to divide that of the numerator.

The result will be

1×(24+x) - 3×(1-2x) = 0

Clear brackets

24 + x-3+6x= 0

Collect like terms

24-3 +x+6x=0

21+7x=0

7x= -21

Divide both side by 7

X = 3.

PERCENTAGE CHANGES
Percentage form a very good basis for comparing changes in the form of increase or decrease relative to the original values. Suppose commodity A increase I price from N500 to N800and commodity B increases in price from N 20 to N 35 within the same time. The increase in the original price of goods A and B are N 300 and N 15 respectively. This shows that A has increased far more in price than B.
This conclusion is faulty because we did not consider the original prices of the goods to know which one has increased more in price relatively to their original price.
To do this, we say :
Percentage increase in price of A = 300
--------- × 100%
500
= 60%

Percentage increase in price B = 15
----- × 100%
20
= 75%
This shows that the rate of increase is actually higher in B than in A . This is because for each N1. 00 in A , there is an increase of 60 Kobo but for each N 1.00 in B , there is an increase of 75 Kobo.
Example 1:
By what must a number be multiplied to
(a) increase it by 20% ( b) decrease it by 15% ?
SOLUTION
(a) The ratio of the new value to the old is 120: 100
Thus , the new value = 120
--------
100
times the old value.
= 6
---- or 1.2 times the old
5
value.

(b) The ratio of the new value to the old is 85 : 100
Thus , the new value = 85
------
100
times the old value.
or 17
---- or 8.5 times the old
20
value.

INCREASE AND DECREASE OF A QUANTITY IN A GIVEN RATIO .

If the length of time for carrying out an assignment is increased from 40 minutes to 1 hour , the ratio of the new time to the old time 60:40 = 3:2. So we say that the time has increased in the ratio 3:2
That is, the new time is 3/2 times the old time.
On the other hand , if the length of time is reduced from 54 minutes to 45 minutes, the ratio of the new time to the old time is 45:54 = 5:6, and we say that the time has decreased in the ratio 5:6. That is, the new time is 5/6 times the old time.
The fraction 5/6 by which the old time , 54 minutes must be multiplied to give the new time 45 minutes is called the multiplying factor. i.e
New quantity
---------------------- =
Old quantity
Multiplying factor.
The multiplying factor is greater than 1 when there is an increase in the ratio and the new quantity is greater than the old quantity; it is less than 1 when there is a decrease in the ratio and the new quantity is less than the old quantity.

Example 1:
Increase N 3.50 in the ratio 10: 7
Solution
New amount = N 3.5 × 10
-----
7
= N 5. 00
Example 2:
Decrease 30kg in the ratio 5:6
Decreased weight = 30 × 5
---
6
= 25kg.
Example 3:
In what ratio must a length of 20m be increased to become 35m?
The ratio 35m: 20 = 35:20 =7:4
Therefore, if 20m is increased in the ratio 7:4 , it becomes 35m.
Check: 20 × 7
---- m = 35m.
4

MORE EXAMPLES ON RATIO
Example 3:
Wole, Chima and Bako agreed to share profits ( or losses) in the ratio of 4:3:6. Divide their profit of N 78,000 among them.

SOLUTION
The ratio 4:3:6 indicates that the profit will be shared into 4+3+6= 13 shares. Wole will receive 4 of the 13 shares , Chima will receive 3 of the 13 shares while Bako will receive 6 of the 13 shares.
Wole receives 4/13 of the profit = 4
---× N78, 000
13
= N24,000

Chima receives 3/13 of the profit = 3
----× N 78, 000
13
= N18, 000

Bako receives 6/13 of the profit = 6
---× N 78, 000
13
= N 36, 000.

MORE EXAMPLES ON RATIO
Example 2:
Two brothers Peter and Paul were instructed by their parents to share a Pocket money of N400 in the ratio of 3 : 5 . How much does each receive?
SOLUTION
If we treat the part of the ratio as share :
Peter receives 3 shares while Paul receives 5 shares.
Total number of shares is 3+5=8
Amount in 8 shares = N400
Amount in 1 share = N 400
----------- =
8
N50
Amount in 3 shares = N50 × 3 = N 150
Amount in 5 shares = N50 × 5 = N250
Therefore , Peter receives N150 while Paul receives N250.

Example 3:
Partners Wole, Chima and Bako agreed to share profits (or losses) in the ratio of 4:3:6. Divide their profit of N78, 000 among them.
SOLUTION
The ratio 4:3:6 indicates that the profit will be shared into 4 + 3 + 6 = 13 shares. Wole will receive 4 of the 13 shares, Chima will receive 3 and Bako will receive 6 of the 13 shares.

Wole receives 4
--- of the profits =
13
4
--- × N78, 000
13
= N24, 000

Chima receives 3
--- of the profit =
13
3
----- × N78, 000 = N18, 000
13

Bako receives 6
--- of the profit =
13
6
---- × N 78, 000 = N36, 000
13

RATIO
Example 1:
(a) The ratio of the cost price to
the selling price of an
article is 4:5. If the article
cost N200, find the selling
Price.
(b) The ratio of the cost price
to the selling price of an
article is 2:3. If the article
was sold N 600, what was
the cost price.

SOLUTION
(a) Ratio of cost price to the selling price is 4:5.
That is , the selling price is 5
----
4
of the cost price
5 5
---- of N 200 = --- × N200
4 4

= N250

(b) The ratio of cost price to selling price is 2:3
That is , the cost price is 2
--- of
3
the selling price.

2 2
-- of N600 = ----- × N600
3 3

= N400.

RATIO
A ratio is a way of comparing numbers or the relationship which one quantity bears to another quantity of the same kind with regard to magnitude.
NOTATION
Ratios can be written in any of the following three ways:
(a) 5 to 7
(b) 5 : 7
(c) 5
----
7
Method (c) indicates that all ratio may be looked at as fractions and may therefore, be multiplied or divided by the same number without altering it's value. Usually, they are expressed as a whole numbers in their lowest terms. Thus 25 : 125 =
1
1:5 or --
5
And 1 1 1 4 4
--- : -- = -- × --- = ---
2 4 2 1 2
When you divide 4 with 2, it becomes 2
---
1
That is 2 : 1

If the prices of two books X and Y are N75 and N200 respectively, then
Price of X N75 3
---------------- = -------- = ----- and
Price of Y N200 8

Price of Y N200 8
---------------- = ---------- = ----
Price of X N75 3
These ratios are written as
Price of X : price of Y = 3 : 8 and
Price of Y : Price of X = 8:3

Conversely, saying that the ratio of the price of X to the price of Y is 3 : 8 means that the price of X is
3
-- of the price of Y, and that the price of Y is
8
-- of the price of X .
3