Ratio and Proportion

Ratio and Proportion: 200: X:: 100: 0.8 (Enter numeric value only.)

The correct answer and Explanation is :

To solve the ratio and proportion problem ( 200:X::100:0.8 ), we can use the concept of cross-multiplication. Here’s the step-by-step solution:

1. Set up the proportion:
   [
   \frac{200}{X} = \frac{100}{0.8}
   ]
2. Cross-multiply:
   [
   200 \times 0.8 = 100 \times X
   ]
3. Calculate ( 200 \times 0.8 ):
   [
   200 \times 0.8 = 160
   ]
4. Set up the equation from cross-multiplication:
   [
   160 = 100 \times X
   ]
5. Solve for ( X ):
   [
   X = \frac{160}{100} = 1.6
   ]

Thus, the value of ( X ) is ( 1.6 ).

Explanation:

In ratio and proportion problems, we use the principle that the cross products of the ratios must be equal. Here’s a more detailed breakdown:

1. Understanding Ratios: Ratios compare two quantities, showing how many times one value is contained in another. The proportion ( 200:X::100:0.8 ) means that the ratio of 200 to X is the same as the ratio of 100 to 0.8.
2. Setting Up the Proportion: To find the unknown value ( X ), we set up a proportion equation where the cross products of the ratios are equal. This means multiplying the numerator of one ratio by the denominator of the other ratio and vice versa.
3. Cross-Multiplication: This technique simplifies solving proportions by creating a straightforward equation. Here, ( 200 \times 0.8 ) equals ( 100 \times X ), which is then solved for ( X ).
4. Solving for ( X ): After calculating ( 200 \times 0.8 = 160 ), we set up the equation ( 160 = 100 \times X ). By isolating ( X ) by dividing both sides by 100, we find ( X = 1.6 ).

This method ensures the ratios are proportionally consistent, making it easier to solve for unknown values in proportional relationships.