Question

A jar of marbles contains two sizes of marbles: normal and jumbo. There are 84 normal size marbles. If \(\frac{2}{9}\) of the marbles are jumbo size, then how many marbles total are in the jar? A. 19 B. 103 C. 108 D. 378

Short answer

The total number of marbles in the jar is 108. The correct option is C. 108.

Step-by-step solution

Find the number of jumbo marbles

Let x be the total number of marbles in the jar. According to the problem, \(\frac{2}{9}\) of the marbles are jumbo size. So, the number of jumbo marbles can be written as \(\frac{2x}{9}\). We know there are \(84\) normal marbles, so we can write the equation as: \[\frac{7}{9}x = 84\]

Solve the equation to find the total number of marbles

Now we need to solve the equation for x, which will give us the total number of marbles in the jar. \[\frac{7}{9}x = 84\] To solve for x, multiply both sides by \(\frac{9}{7}\): \[x = \frac{9}{7} \cdot 84\] \[x = 9 \cdot 12\] \[x = 108\] So, there are 108 marbles in total in the jar.

Choose the correct option

Now that we have found the total number of marbles, we can compare our answer to the given options: A. 19 B. 103 C. 108 D. 378 Since 108 is among the choices, the correct answer is: C. 108

Key concepts

Ratio and Proportion

Understanding ratio and proportion is essential when facing problems that involve comparing quantities or scaling numbers in proportion to one another. A ratio simply compares two quantities, often in the form 'A to B' or as a fraction like \( \frac{A}{B} \). When we talk about proportion, we are dealing with an equation that shows two ratios are equivalent.

For example, if we have a scale model where 1 inch represents 5 miles, we have a ratio of 1 to 5. If our model is 2 inches long, then the actual distance represented is in proportion to the scale, which in this case would be 10 miles. This ratio (scale model to actual size) remains constant.

In our exercise, we use proportion to determine the total number of marbles in a jar. We know that the ratio of jumbo to normal size marbles is \( \frac{2}{9} \). Using this, we can set up a proportional relationship (\