The Prices Of Two Radios Are In The Ratio $x: Y$. When The Prices Are Both Increased By £ 40 £40 £40 , The Ratio Becomes 5 : 3 5: 3 5 : 3 . When The Prices Are Both Reduced By £ 5 £5 £5 , The Ratio Becomes 2 : 1 2: 1 2 : 1 .Express The Ratio

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Introduction

In this article, we will explore the concept of ratios and how they can be used to solve problems involving the prices of two radios. We will use the given information to set up a system of equations and solve for the ratio of the prices of the two radios.

The Problem

The prices of two radios are in the ratio $x: y$. When the prices are both increased by £40, the ratio becomes 5:3. When the prices are both reduced by £5, the ratio becomes 2:1.

Setting Up the Equations

Let's start by setting up the equations based on the given information. We know that the ratio of the prices of the two radios is $x: y$. When the prices are both increased by £40, the ratio becomes 5:3, so we can set up the equation:

x+40y+40=53\frac{x+40}{y+40} = \frac{5}{3}

Similarly, when the prices are both reduced by £5, the ratio becomes 2:1, so we can set up the equation:

x−5y−5=21\frac{x-5}{y-5} = \frac{2}{1}

Solving the Equations

Now that we have set up the equations, we can solve them to find the ratio of the prices of the two radios. We can start by cross-multiplying the first equation:

3(x+40)=5(y+40)3(x+40) = 5(y+40)

Expanding and simplifying the equation, we get:

3x+120=5y+2003x + 120 = 5y + 200

We can also cross-multiply the second equation:

y−5=2(x−5)y-5 = 2(x-5)

Expanding and simplifying the equation, we get:

y=2x−10y = 2x - 10

Substituting and Solving

Now that we have the two equations, we can substitute the expression for y from the second equation into the first equation:

3x+120=5(2x−10)+2003x + 120 = 5(2x - 10) + 200

Expanding and simplifying the equation, we get:

3x+120=10x−50+2003x + 120 = 10x - 50 + 200

Combine like terms:

3x+120=10x+1503x + 120 = 10x + 150

Subtract 3x from both sides:

120=7x+150120 = 7x + 150

Subtract 150 from both sides:

−30=7x-30 = 7x

Divide both sides by 7:

x=−307x = -\frac{30}{7}

Finding the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We can use the second equation:

y=2x−10y = 2x - 10

Substituting x = -30/7, we get:

y=2(−307)−10y = 2(-\frac{30}{7}) - 10

Simplifying the equation, we get:

y=−607−10y = -\frac{60}{7} - 10

y=−607−707y = -\frac{60}{7} - \frac{70}{7}

y=−1307y = -\frac{130}{7}

The Ratio

Now that we have the values of x and y, we can express the ratio of the prices of the two radios as:

x:y=−307:−1307x: y = -\frac{30}{7}: -\frac{130}{7}

We can simplify the ratio by dividing both numbers by their greatest common divisor, which is 1:

x:y=−307:−1307x: y = -\frac{30}{7}: -\frac{130}{7}

Conclusion

In this article, we used the given information to set up a system of equations and solve for the ratio of the prices of the two radios. We found that the ratio is $-\frac30}{7} -\frac{130{7}$.

Final Answer

The final answer is $-\frac30}{7} -\frac{130{7}$.

Introduction

In our previous article, we explored the concept of ratios and how they can be used to solve problems involving the prices of two radios. We used the given information to set up a system of equations and solve for the ratio of the prices of the two radios. In this article, we will answer some of the most frequently asked questions about the problem.

Q&A

Q: What is the ratio of the prices of the two radios?

A: The ratio of the prices of the two radios is $-\frac30}{7} -\frac{130{7}$.

Q: How did you solve the problem?

A: We used the given information to set up a system of equations and solve for the ratio of the prices of the two radios. We started by setting up two equations based on the given information and then solved them to find the ratio.

Q: What is the difference between the prices of the two radios?

A: To find the difference between the prices of the two radios, we can subtract the smaller price from the larger price:

−1307−(−307)=−1307+307=−1007-\frac{130}{7} - (-\frac{30}{7}) = -\frac{130}{7} + \frac{30}{7} = -\frac{100}{7}

So, the difference between the prices of the two radios is $-\frac{100}{7}$.

Q: What is the ratio of the prices of the two radios in simplest form?

A: The ratio of the prices of the two radios is already in simplest form:

−307:−1307-\frac{30}{7}: -\frac{130}{7}

Q: Can you explain the concept of ratios in more detail?

A: A ratio is a way of comparing two or more numbers. It is a fraction that shows the relationship between two or more quantities. In this problem, we used the ratio to compare the prices of two radios.

Q: How do you know that the ratio is $-\frac30}{7} -\frac{130{7}$?

A: We solved the system of equations to find the ratio of the prices of the two radios. We started by setting up two equations based on the given information and then solved them to find the ratio.

Q: Can you give an example of how to use ratios in real-life situations?

A: Ratios are used in many real-life situations, such as cooking, building, and finance. For example, a recipe for making cookies might call for a ratio of 2:3 of sugar to flour. This means that for every 2 parts of sugar, you need 3 parts of flour.

Conclusion

In this article, we answered some of the most frequently asked questions about the problem of the prices of two radios in the ratio $x: y$. We explained the concept of ratios and how they can be used to solve problems involving the prices of two radios.

Final Answer

The final answer is $-\frac30}{7} -\frac{130{7}$.