Question

1. write and solve an equation to find the value of x. 4.8, 1.6, 5.2, x; mean is 4.2

a. x = 5.2
b. x = 4.2
c. x = 11.6
d. x = 4

Explanation:

Step1: Recall the mean formula

The mean (average) of a set of numbers is calculated by the sum of the numbers divided by the count of the numbers. The formula for the mean \(\bar{x}\) of a set \(\{x_1, x_2, \dots, x_n\}\) is \(\bar{x}=\frac{x_1 + x_2+\dots+x_n}{n}\). Here, we have the numbers \(4.8\), \(1.6\), \(5.2\), and \(x\), so \(n = 4\) and \(\bar{x}=4.2\).

Step2: Set up the equation

Substitute the known values into the mean formula: \(4.2=\frac{4.8 + 1.6+5.2+x}{4}\)

Step3: Multiply both sides by 4

To eliminate the denominator, multiply both sides of the equation by \(4\): \(4.2\times4=4.8 + 1.6+5.2+x\)
Calculate the left - hand side: \(16.8=4.8 + 1.6+5.2+x\)

Step4: Simplify the right - hand side

First, add \(4.8\), \(1.6\), and \(5.2\): \(4.8+1.6 = 6.4\), then \(6.4 + 5.2=11.6\). So the equation becomes \(16.8=11.6+x\)

Step5: Solve for x

Subtract \(11.6\) from both sides of the equation: \(x=16.8 - 11.6\)
Calculate the result: \(x = 5.2\)? Wait, no, wait. Wait, \(16.8-11.6 = 5.2\)? Wait, no, let's recalculate. Wait, \(4.8+1.6 = 6.4\), \(6.4 + 5.2=11.6\). Then \(16.8-11.6 = 5.2\)? But wait, let's check again. Wait, maybe I made a mistake. Wait, the numbers are \(4.8\), \(1.6\), \(5.2\), \(x\). The mean is \(4.2\). So sum of numbers is \(4.2\times4=16.8\). Sum of \(4.8 + 1.6+5.2=4.8+1.6 = 6.4\), \(6.4 + 5.2 = 11.6\). Then \(x=16.8 - 11.6=5.2\)? But option a is \(x = 5.2\). Wait, but let's check the mean again. If \(x = 5.2\), the numbers are \(4.8\), \(1.6\), \(5.2\), \(5.2\). The sum is \(4.8+1.6+5.2+5.2=16.8\), and the mean is \(16.8\div4 = 4.2\), which matches. Wait, but let's check the options again. Option a is \(x = 5.2\). Wait, but maybe I made a mistake in calculation. Wait, let's re - do the steps:

1. Mean formula: \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), \(n = 4\), \(\bar{x}=4.2\), \(x_1 = 4.8\), \(x_2=1.6\), \(x_3 = 5.2\), \(x_4=x\)
2. Equation: \(4.2=\frac{4.8 + 1.6+5.2+x}{4}\)
3. Multiply both sides by 4: \(4.2\times4=4.8 + 1.6+5.2+x\)

• \(4.2\times4 = 16.8\)
• \(4.8+1.6+5.2=11.6\)

1. Then \(16.8=11.6+x\)
2. Subtract \(11.6\) from both sides: \(x=16.8 - 11.6 = 5.2\)

Answer:

a. \(X = 5.2\)