Question

In the air talent show in the eve of independence day 90 aircraft (only Sukhoi and Mig-21) participated. It is known that for every one hour flight a Sukhoi and a Mig-21 consume 2 litres and 3 litres of fuel respectively. The price of fuel for Sukhoi is \(\$ 10\) per litre and price for the fuel of Mig-21 is \(\$ 15\) per litre. Thus all the aircrafts accounted \(\$ 3050\) for the fuel in one hour show. The number of Sukhoi planes is : (a) 20 (b) 30 (c) 40 (d) 50

Short answer

Answer: 40 Sukhoi planes participated in the air talent show.

Step-by-step solution

Define variables

Let \(x\) be the number of Sukhoi planes and \(y\) be the number of Mig-21 planes. Since there are a total of 90 aircraft participating in the show, we can write the first equation as: $$x + y = 90$$

Create another equation using fuel price and consumption information

We're given that Sukhoi planes consume 2 litres of fuel per hour at a cost of \(10 per litre, and Mig-21 planes consume 3 litres of fuel per hour at a cost of \)15 per litre. The total fuel cost for all planes after one hour is $3050. We can write the second equation as: \(2x * 10 + 3y * 15 = 3050\)

Simplify the second equation

Now, we simplify the above equation by multiplying the constants: $$20x + 45y = 3050$$

Solve the system of equations

We can use the first equation to find an expression for either \(x\) or \(y\) and substitute it in the second equation to solve for the other variable. From equation one we can find an expression for \(y\): $$y = 90 - x$$ Plug this into the second equation: $$20x + 45(90-x) = 3050$$

Solve for x

Now solve for \(x\): $$20x + 4050 - 45x = 3050$$ $$-25x = -1000$$ $$x = 40$$

Find the number of Sukhoi planes

Since \(x\) represents the number of Sukhoi planes, we have found that there are 40 Sukhoi planes in the air talent show. The answer is (c) 40.

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two algebraic expressions. These expressions can include numbers, variables (such as x and y), and operations like addition, subtraction, multiplication, and division. In the context of the exercise, we use algebraic equations to represent the relationship between the number of aircraft and the costs associated with their fuel consumption.

For instance, the equation x + y = 90 is a simple linear algebraic equation where x and y symbolize the number of Sukhoi and Mig-21 planes, respectively. This equation reflects the total count of aircraft. Similarly, 20x + 45y = 3050 is another linear algebraic equation derived from the given information on fuel consumption rates and costs. Algebraic equations are the foundation of solving problems that require an affirmative solution to an unknown variable.

System of Equations

A system of equations consists of two or more equations that share common variables and are solved together. The goal is to find the values of the variables that make all the equations true simultaneously. In the provided example, the system is made up of the equations x + y = 90 and 20x + 45y = 3050. These equations are solved concurrently to find the values of x (the number of Sukhoi planes) and y (the number of Mig-21 planes).

To solve the system, one can use methods like substitution, elimination, or graphing. The substitution method was used in this particular case, where x was expressed in terms of y using the first equation, and then substituted into the second equation. This strategy simplifies the process, making it easier to solve for one variable at a time.

Fuel Consumption Problem

A fuel consumption problem is a type of word problem that deals with the rate at which vehicles consume fuel and the associated costs. This type of problem typically involves creating an equation or system of equations based on given conditions to determine one or more unknowns, such as the amount of fuel used, the total cost, or the number of vehicles.

In our exercise, the problem revolves around calculating the number of two types of aircraft based on their fuel consumption rates and the total fuel cost for an air show. The creation of a system of equations allows us to use known values (fuel rates and costs) to determine the unknown number of each type of plane.

Word Problem Solving

Word problem solving involves translating a real-world situation described in words into a mathematical model, such as an equation or a system of equations. The key steps in solving a word problem include reading the problem carefully, identifying and defining variables, writing down the equations representing the problem scenario, and solving the equations to find the values of the variables.

In the context of our exercise, the word problem describes an air show with specific conditions on fuel consumption and costs. The solver needs to define variables (x for Sukhoi and y for Mig-21), convert the text into equations (total aircrafts and total fuel cost), and use algebraic methods to find the solution. This approach turns a verbal description into a solvable mathematical challenge.