S3TN Solving Fractional Equations and Others

Solving Fractional Quadratic Equations

Cross-Multiplying =>

            a = c     =>      x bd  a  =   c   x bd    =>   ad = bc

            b    d                        b       d

[[ The Equation must be ax²+ bx + c = 0 before factorising ]]

Example 

solve    6   = x + 3

          x + 4

            6      =    x + 3 

         x + 4

 6 = (x + 3)(x + 4)         (Step1:  Cross-Multiply)

   6 = x²+ 7x + 12          (Step2: Expand to form a quadratic equation)

   x²+ 7x + 12 – 6 = 0

   x²+ 7x  + 6 = 0

  (x + 1)(x + 6) = 0         (Step3: Factorise and Solve for x)

   x + 1 = 0 or x + 6 = 0

   x = -1 or x = -6

Example

Solve      1       +    2       = 5

            x -2        x – 3 

              (x – 3)  + 2 x (x – 2)   = 5        (Step1: LCM the denominator)

        (x - 3)(x -2)  (x – 2)(x – 3)

            x – 3  + 2x – 4  =  5                   (Step 2 : Combine to single fraction)

              (x – 2)(x -3)

         x + 2x – 3 – 4 = 5        

           (x – 2)(x -3)

               3x – 7       = 5

         (x – 2)(x – 3)

          3x – 7 = 5(x – 2)(x – 3).              (Step 3 : Cross-Multiply and Expand)

          3x – 7 = 5(x²– 5x + 6)

          3x – 7 = 5x²– 5x + 6

          5x²– 5x + 6 – 3x + 7 = 0. 

          5x²-8x + 13 = 0                           (Step 4: Factorise and Solve for x)                                       

          (5x + 13)(x + 1) = 0                                           5x   \ /    13

     5x + 13 = 0 or x + 1 = 0                                           x    / \     1  

           x = -13/5 or x = -1

Solving Equations by equating Coefficients

Example

Find the value of A, B and C

2x⁴ - 13x³ + 19x² + 5x + 1 = (x - 4)(Ax + 1)(x² + Bx + 1) + C

Tips : 
1. Such questions are usually solved by equating the coefficient values
2. If there are more than 2 brackets, expands in pairs
3. Solve the unit value first, and then the highest powers of X

Step 1 - Expand brackets (if needed)

   2x⁴ - 13x³ + 19x² + 5x + 1 = (x - 4)(Ax + 1)(x² + Bx + 1) + C

   2x⁴ - 13x³ + 19x² + 5x + 1 = (x - 4)(Ax + 1)(x² + Bx + 1) + C [Expands 1 pair]

                    (x - 4) (Ax + 1) = (Ax² + (1 - 4)x - 4) (x² + Bx + 1) + C

  2x⁴ - 13x³ + 19x² + 5x + 1 = (Ax² + (1 - 4A)x - 4)(x² + Bx + 1) + C

Step 2 - extract  and solve from comparing coefficients 

  Extract coefficient  of x4.   [=> x² x x² , x³ x x , x⁴ x unit value ]

      2x⁴ - 13x³ + 19x² + 5x + 1 = (Ax² + (1 - 4A)x - 4)(x² + Bx + 1) + C

                   2x⁴  =  Ax² x x²

    A = 2  [Equating x4 coefficient]

    

  Extract and equating coefficient of unit value

    1 = (-4 x 1) + C

C = 1 + 4

C = 5

Step 3 - Substitute values and find remaining unknown

     2x⁴ - 13x³ + 19x² + 5x + 1 = (2x² -7x - 4)(x² + Bx + 1) + 5

  Extract and equating x coefficients

5x = -7x x 1 - 4Bx

             5 = -7 - 4B

  4B = -12

                B = -3 

A = 2, B = -3, C = 5

N2-G123 Ratios

Ratio compares two or more quantities.

            Symbol of ratio is :     

Ratio of quantity A to quantity B s written as     

                  A : B                                                    

** Ratio has no unit of measure

Example

Write the ratio 200:  250 in its simplest form

                  200 : 250

                  20÷5 : 25 ÷ 5

                      4 : 5

                  

                  a : b: c => a: b , b : c

Equivalent Ratios

Example

                       4 : 10 

                = 4 _÷2 : 10_÷2

                =     2 : 5

4 : 10 and 2 : 5 are equivalent ratios.

Ratio in its simplest form

Example

Write 8 : 12: 36 in its simplest form

Step: Find the common factors for the given numbers and reduce

            8 _÷4 : 12_÷4 : 36_÷4

            2 : 3 : 9

       

Example: Write the ratio 200:  250 in its simplest form

                    200 : 250

                  20÷5 : 25 ÷ 5

                        4 : 5

Example

The ratio of the number of tables to the number of chairs is 1 : 4 in the restaurant. There are 6 tables, how many chairs are there?

Step 1 : Write the ratio and number of tables

T  C

1 : 4

Step 2 : Write the number of tables on both side of the ratio and multiply

T   C

1  : 4

                    x 4   x4

= 4  : 16

Step3 : Answer 

There are 16 chairs

Ratio with different Unit of Measure (UOM)

The UOM must be the same when comparing two of more quantities

Example:

John has $1 and Sally has 80cts as their daily allowance. What is the ratio of their allowance?

Step 1 : Convert the amount to the same unit (of the 'smaller' UOM)

$1 = 100cts

Step 2 :  Write the ratio and reduce the the lowest term

100 : 80

₅  10 : 8 ₄

      5 : 4

Step 3 : Write the Answer

The ration is 5 : 4

Ratio involving Fractions and Decimals

Example

Express the following ratios in its simplest form.

  1.2 : 3 

Step 1 : Multiply 10 to both side (to remove the decimal)

x10.  1.2. :   3 x 10

12 :  30

Step 2 : Reduce the simplest term

<< divide by 6 for both side >>

    2    12 :  30     5        

The ratio is 2 :  5

Example

Express the following ratios in its simplest form.

b.         1 ¾.  :    2 ½    

Step 1: Change both to improper fractions

    7  :  5

             4     2

Step 2 :  Make both side to have the same denominator

<< multiply 2 , by 2 >>

    7  :  5  x 2

             4     2  x 2

    7  :  10  

             4      4 

Step 3 : Write the numerators as the ratio

7 : 10

Connecting Ratio with Fraction

For a ratio   A : B, the 'total number of parts' = A + B

Thus in fraction form,    A's part is A / A+B

              B's part is B / A+B 

Example

There are 10 girls and 15 boys.

The ratio of the number of girls to the number of boys = 

10 : 15    or    2   :   3

The total part = 2 + 3 = 5

=>  Number of girls = 2/3 of number of boys

=>  Number of girls = 2/5 of the total

=>  Number of boys = 3/2 of number of girls

=>  Number of boys = 3/5 of the total.

Example

Abby and Bobby have 10 oranges altogether. 

The oranges are to be divided in the ratio of 3 : 2

How many oranges will each of them get?

Step 1: Find the total parts 

Total = 3 + 2 = 5

Step 2 : Compute their fraction's

  Abby = 3/5 , Bobby = 2/5

Step 3 : Compute by multiplying total quantity with the fractions

Abby has 3/5 x 10 = 6 oranges

Bobby has 2/5 x 10 = 4 oranges

Practice

1.  Express the following ratios in its simplest form.

(a) 4 :  20 [1 : 5]

(b) 35 : 14 [7 : 2] 

(c)    64 : 56 : 16 [8 : 7 : 2] 

2.   Express the following ratios in its simplest form.

a.   ⅚ kg :  2 kg                        [5 : 12 ]

b.  1.6 cm : 4 mm                     [4 : 1 ]

c.  1 ½ hr : 20 mins                  [9 : 2 ]

d.  250 ml : 1 litre                     [1 : 4 ]

3.  Mary and Andy share a pizza in the ratio of 1 : 2. Andy gets 4 slices of the pizza. How many slices does Mary get? [2]

4.  Alice, Betty and Coby share their marbles in the ratio of 2 : 3 : 5. Find the amount each gets for each of the marbles.

a.  420 (b) 70                   [a. 84, 126, 210     b. 14, 21, 35]