FORMULAE
A formula is a relation between certain quantities, expressed with the help of variables and mathematical symbols.
Example.1) When a number is multiplied by 5 and 12 is deducted from the product, the result is 7 more than thrice the number, if we denote the number ‘x’, we can represent this statement by the formula, 5x – 12 = 3x + 7
Example.2) The volume (V) of a cuboid is the product of its length (l), breadth (b), and height (h). This can be expressed by the formula V = l X b X h.
Example.3) The area (A) of a rectangle is the product of its length (l) and its breadth (b). This can be expressed as A = l X b.
Process of Framing a Formula –
Step.1) Use variables such as a, b, c, d, w, x, y, z, etc. for the quantities for which you want to frame the formula. Certain symbols are traditionally used to denote certain variables.
Examples. A) ‘V’ is used for volume, ‘A’ for area, ‘l’ for length, ‘b’ for breadth, ‘h’ for height, and ‘S’ for surface area in mensuration.
B) ‘v’ & ‘u’ are used for speed, ‘s’ for distance, and ‘t’ for time in physics.
C) ‘P’ is used for principal, ‘A’ for amount, ‘R’ for rate, ‘I’ for interest, and ‘T’ for time in arithmetic.
D) Use the rules or conditions relevant to the context to establish a relationship between the variables.
Example.1) Richard is 25 years older than his son Donald, after 10 years his age will be 7 years more than thrice the age of Donald. Frame a formula for these statements.
Ans.) Let us assume the age of Donald is ‘x’ years
So, as per the given condition, the age of Richard would be = x + 25
After 10 years the age of Donald would be = x + 10
And, After 10 years the age of Richard would be = ( x + 25 ) + 10
= x + 35
Now, as per the given condition, the required equation should be –
x + 35 = 3 ( x + 10 ) + 7
above equation is the required formula. (Ans.)
Example.2) In a three-digit number, the digit in the tens place exceeds the digit in the units place by 7, and reduced the hundred place by 4, write the formula for the number.
Ans.) Let us assume that, the digit in the units place be ‘x’
Then the digit in the tens place = x + 7
And, the digit in the hundreds place = x - 4
So, the number would be = 100 X digit in hundreds place + 10 X digit in tens place + digit in unit place
= 100.( x – 4 ) + 10.( x + 7 ) + x
= 100x – 400 + 10x + 70 + x = ( 100 + 10 + 1 ).x – ( 400 – 70 )
= 111x – 330 (Ans.)
Example.3) In all, 250 tickets were sold for a charity show. Adult tickets cost $ 20 each and student tickets cost $ 10 each. If the number of adult tickets sold was x, construct a formula for the income ‘I’ from the show.
Ans.) As per the given situation, total of 250 tickets were sold and the number of the adult ticket sold was ‘x’
So, the number of student ticket was 250 – x
If the cost of adult ticket was $ 20 each and the number of adult ticket sold was ‘x’, so the total value came from the only adult ticket was = 20x
If the cost of a student ticket is $ 10 and the number of students were 250 – x, then the total value comes from only student ticket was = 10 ( 250 – x )
Now, if ‘I’ was the total income from the show –
I = 20x + 10.( 250 – x )
= 20x + 2500 – 10x = 10x + 2500 (Ans.)
Example.4) Steve has an average score of 58 runs in ‘x’ innings and an average of 65 runs in ‘y’ innings, find the average score ‘A’ for ‘x’ & ‘y’ innings.
Ans.) As per the given condition, Steve has an average score of 58 runs in ‘x’ innings
So, the total runs scored in ‘x’ innings is = 58 X x = 58x
And, Steve has an average score of 65 runs in ‘y’ innings
So, the total number scored in ‘y’ innings is = 65y
so, total runs scored by Steve = ( 58x + 65y )
Total innings played by Steve is = ( x + y ) 58x + 65y
So, the equation of finding the average score is A = ------------ (Ans.)
x + y
Example.5) Pollard earns a profit of $ 300 by selling 15 toys at the rate of $ x per toy. If the cost price of each toy is $ 40, frame a formula for the profit.
Ans.) as per the given condition, the cost price is $ 40 for the 15 toys each.
So, the total cost price of all the toys is = $ 40 X 15 = $ 600
Now, Pollard sold out 15 toys at the cost of $ x per toys, so the total selling price is $ 15x
Pollard has earned a profit of $ 200
Now, as per the given condition, the desired equation would be –
15x – 600 = 300
The above equation is a required formula. (Ans.)