Whenever we use ‘mathematical statements’ to convert verbal information with an aim to describe relationships between things that vary with circumstances and time etc, we end up writing verbal information into either a mathematical expression or an equation (or even an inequation). Essentially, we convert verbal information into mathematical language using symbols etc.
When converting word problems to equations, certain “key words” tell you what kind of operations to use: addition, multiplication, subtraction, and/ or division. The table below shows some common phrases for the operation of addition, subtraction and product.
| Word | Operation | Example | As an equation |
|---|---|---|---|
| sum | addition | The sum of my age and 10 equals 27. | y + 10 = 27 |
| difference | subtraction | The difference between my age and my younger sister’s age, who is 11 years old, is 5 years. | y - 11 = 5 |
| product | multiplication | The product of my age and 14 is 168. | y × 14 = 168 |
| times | multiplication | Three times my age is 60. | 3 × y = 60 |
| less than | subtraction | Seven less than my age equals 32. | y - 7 = 32 |
| total | addition | The total of my pocket change and Rs 20 is Rs. 22.75 | y + 20 = 22.75 |
| more than | addition | Eleven more than my age equals 43. | 11 + y = 43 |
Examples: Show mathematically:
a. If 29 be divided by 6, the quotient is 4, and the remainder is 5.
Mathematically, this can be shown as:
b. In a class of x students, y play cricket, z play hockey, and the rest are spectators. What is the excess of students who play either game over the students who are spectators?
To show this mathematically, we first need to find the number of students who play a game and then find the number of students who are spectators followed by showing the difference between the two.
Number of students who play a game = y+z
Number of students who are spectators = x-(y+z)
The excess of students who play a
game over the students who are spectators = (y+Z)-[x-(y+z)]
Simplifying the above the excess can be given = 2y+2z-x
In the above example — x, y and z are called variables.
From the above examples it is apparent that to understand and master this conversion of verbal information to usable mathematical language, implicitly includes that the candidate will understand both the verbal part as well as will know the mathematical part to be able to accomplish the task.
However, not all equations are this simple! Try this one yourself before you look up the solution.
Determine the price P, of a book that depends on the quantity Q, of the books purchased. Let the base price for purchasing one book be Rs.150. Also, given is that for each additional book purchased, the cost per book is reduced by Rs. 2. Write an equation for P in terms of Q.
The following equation represents the relationship between the price PQ of the Qth book purchased by the equation:
PQ = (–2) (Q – 1) + 150, where Q = 1, 2, 3, 4…
P1=50
p2=148
p3=146 and so on
Here, ∑PQ will give the total price for all the books purchased.
The equations show that two expressions are equivalent. In order to understand the relationship being represented, we will need to be able to solve the equations used to describe these relationships.
Solving equations (with one or two or more variables) is discussed in another article.
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