Question

marnie solved the proportion $\frac{150}{170}=\frac{x}{510}$ to find the value of $x$ in the enlarged parallelogram.
what is the value of $x$?
$\bigcirc x = 255$
$\bigcirc x = 300$
$\bigcirc x = 340$
$\bigcirc x = 450$

Explanation:

Step1: Cross - multiply the proportion

Given the proportion \(\frac{150}{170}=\frac{x}{510}\), we cross - multiply. Cross - multiplying means that we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the numerator of the second fraction multiplied by the denominator of the first fraction. So we get \(170x = 150\times510\).

Step2: Calculate the right - hand side

First, calculate \(150\times510\). \(150\times510 = 150\times(500 + 10)=150\times500+150\times10 = 75000+1500 = 76500\). So the equation becomes \(170x=76500\).

Step3: Solve for x

To solve for \(x\), we divide both sides of the equation \(170x = 76500\) by 170. So \(x=\frac{76500}{170}\). Simplify the fraction: \(\frac{76500\div170}{170\div170}=\frac{450}{1}\)? Wait, no, wait. Wait, \(76500\div170\): Let's divide numerator and denominator by 10 first, we get \(\frac{7650}{17}\). And \(17\times450 = 7650\), so \(x = 450\)? Wait, no, wait, that's a mistake. Wait, \(150\times510 = 76500\), \(76500\div170\): \(170\times450=76500\)? Wait, \(170\times400 = 68000\), \(170\times50 = 8500\), \(68000 + 8500=76500\). Wait, but let's check the proportion again. Wait, the original proportion is \(\frac{150}{170}=\frac{x}{510}\). Let's simplify \(\frac{510}{170}=3\). So \(x = 150\times3=450\)? Wait, but the options have \(x = 450\) as an option. Wait, but let's re - check.

Wait, maybe I made a mistake in cross - multiplying. Wait, the proportion is \(\frac{150}{170}=\frac{x}{510}\). Cross - multiplying: \(170x=150\times510\). \(150\times510 = 76500\). Then \(x=\frac{76500}{170}\). Divide numerator and denominator by 10: \(\frac{7650}{17}\). \(17\times450 = 7650\), so \(x = 450\).

Wait, but let's check the ratio of the bases. The base of the small parallelogram is 170, the base of the large one is 510. \(510\div170 = 3\), so the scale factor is 3. Then the height of the small one is 150, so the height of the large one should be \(150\times3 = 450\). So \(x = 450\).

Answer:

\(x = 450\) (the option is \(x = 450\))