Direct and Inverse variation

Variation describes the relation between two or more quantities. e.g. workers and work done, children and noise, entrepreneurs and start ups. More workers means more work done; more children means more noise; more entrepreneurs means more start ups and so on… These are examples of direct variation i.e. if one quantity increases, the other increases proportionally. Then there are quantities that have inverse variation between them e.g. workers and time taken. If there are more workers, time taken to complete a work will be less. DIRECT VARIATION: Formally, let’s say x varies directly with y. If x takes values x1, x2, x3… and y takes values y1, y2, y3 … correspondingly, then x1/y1 = x2/y2 = x3/y3 = Some constant value In other words, ratio of x and y stays the same in different instances. (Notice that this is the same as x1/x2 = y1/y2) It might seem a little cumbersome when put this way but the truth is that direct variation is quite intuitive. A couple of questions will make it clear. Question 1: 20 workmen can make 35 widgets in 5 days. How many workmen are needed to make 105 widgets in 5 days? (A)   7 (B)   20 (C)   25 (D)    60 Solution: Notice that the number of days stays the same so we can ignore it. Now think, how are workmen and widgets related? If the number of workmen increases, the number of widget made also increases proportionally. You need to find the new number of workmen required. The number of widgets has become thrice (105/35 = 3) so number of workmen needed will become thrice as well (remember, the number of workmen will increase in the same proportion). We need 20*3 = 60 workmen Answer (D) The concept of variation is very intuitive. If the number of widgets required doubles, the number of workmen required to make them in the same amount of time will double too. If the number of widgets required becomes one fourth, the number of workmen required to make them in the same amount of time will become one fourth too. A quantity can directly vary with some power of another quantity. Let’s take an example of this scenario too. Question 2: If the ratio of the volumes of two right circular cylinders is given by 64/9, what is the ratio of their radii? (Both the cylinders have the same height) (A)   4/3 (B)   8/3 (C)   16/9 (D)   4/1 Solution: This question involves a little bit of geometry too. The volume of a right circular cylinder is given by Area of base * height i.e. Volume of a right circular cylinder = pi*radius^2 * height So volume varies directly with the square of radius. Va/Vb = 64/9 = Ra^2/Rb^2 Ra/Rb = 8/3 Answer (B) INVERSE VARIATION: The concept of inverse variation is very simple – two quantities x and y vary inversely if increasing one decreases the other proportionally. If x takes values x1, x2, x3… and y takes values y1, y2, y3 … correspondingly, then x1*y1 = x2*y2 = x3*y3 = some constant value This means that if x doubles, y becomes half; if x becomes 1/3, y becomes 3 times etc. In other words, product of x and y stays the same in different instances. Notice that x1/x2 = y2/y1; The ratio of x is inverse of the ratio of y. The concept will become clearer after working on a few examples. Question 1: The price of a diamond varies inversely with the square of the percentage of impurities. The cost of a diamond with 0.02% impurities is $2500. What is the cost of a diamond with 0.05% impurities (keeping everything else constant)? (A)   $400 (B)   $500 (C)   $1000 (D)   $4000 Solution: Price1*(% of Impurities1)^2 = Price2*(% of Impurities2)^2 2500*(.02)^2 = Price2*(.05)^2 Price = $400 Answer (A) The answer is quite intuitive in the sense that if % of impurities in the diamond increases, the price of the diamond decreases. There is an important question type related to inverse variation. It often uses the formula: Total Price = Number of units*Price per unit If, due to budgetary constraints, we need to keep the total money spent on a commodity constant, number of units consumed varies inversely with price per unit. If price per unit increases, we need to reduce the consumption proportionally. Question 2: The cost of fuel increases by 10%. By what % must the consumption of fuel decrease to keep the overall amount spent on the fuel same? (A)   5% (B)   9% (C)   10% (D)   11% Solution: Do you think the answer is 10%? Think again. Total Cost = Number of units*Price per unit If the price per unit increases by 10%, it becomes 11/10 of its original value. To keep the total cost same, you need to multiply number of units by 10/11. i.e. you need to decrease the number of units by 1/11 i.e. 9.09%. In that case, New Total Cost = (10/11)*Number of units*(11/10)*Price per unit This new total cost will be the same as the previous total cost. Answer (B) Let’s look at one more example of the same concept but this one is a little trickier. Question 3: Recently, fuel price has seen a hike of 20%. Mr X is planning to buy a new car with better mileage as compared to his current car. By what % should the new mileage be more than the previous mileage to ensure that Mr X’s total fuel cost stays the same for the month? (assuming the distance traveled every month stays the same) (A)   10% (B)   17% (C)   20% (D)   21% Solution:  The problem here is ‘how is mileage related to fuel price?’ Total fuel cost = Fuel price * Quantity of fuel used Since the ‘total fuel cost’ needs to stay the same, ‘fuel price’ varies inversely with ‘quantity of fuel used’. Quantity of fuel used = Distance traveled/Mileage Distance traveled = Quantity of fuel used*Mileage Since the same distance needs to be traveled, ‘quantity of fuel used’ varies inversely with the ‘mileage’. We see that ‘fuel price’ varies inversely with ‘quantity of fuel used’ and ‘quantity of fuel used’ varies inversely with ‘mileage’. So, if fuel price increases, quantity of fuel used decreases proportionally and if quantity of fuel used decreases, mileage increases proportionally. Hence, if fuel price increases, mileage increases proportionally or we can say that fuel price varies directly with mileage. If fuel price becomes 6/5 (20% increase) of previous fuel price, we need the mileage to become 6/5 of the previous mileage too i.e. mileage should increase by 20% too. Another method is that you can directly plug in the expression for ‘Quantity of fuel used’ in the original equation. Total fuel cost = Fuel price * Distance traveled/Mileage Since ‘total fuel cost’ and ‘distance traveled’ need to stay the same, ‘fuel price’ is directly proportional to ‘mileage’. Answer (C)