Jar contains red, white, and yellow candy pieces in the...

jar contains red, white, and yellow candy pieces in the ratio of $9: 5: 4,$ respectively. When 7 pieces of red candy and 5 pieces of white candy are removed from the jar and 3 pieces of yellow candy are added, the ratio of red to white to yellow becomes 4: 2: 3 . If the jar contains only these three colors of candy, how many pieces were originally in the jar?

jar contains red, white, and yellow candy pieces in the ratio of $9: 5: 4,$
respectively. When 7 pieces of red candy and 5 pieces of white candy are removed from the jar and 3 pieces of yellow candy are added, the ratio
of red to white to yellow becomes 4: 2: 3 . If the jar contains only these three colors of candy, how many pieces were originally in the jar?

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Key Concepts

Ratio

A ratio expresses the relative amounts of different items by showing how many times one value is contained in another. In problems like this, ratios provide a way to understand the relationship between different groups, such as the red, white, and yellow candies.

Scaling Factor

When items are in a particular ratio, each term in the ratio is multiplied by a common scaling factor to obtain the actual quantities. This factor is essential in converting the proportional relationships into concrete numbers.

Adjustments to Ratios

Modifying a set by adding or removing items changes the original ratio. Recognizing how these adjustments affect the proportional distribution is key to setting up the problem correctly, particularly when the resulting ratio is provided after the change.

Systems of Equations

Establishing relationships based on initial and adjusted ratios typically leads to the formulation of a system of equations. Solving these equations simultaneously allows one to find the unknown quantities, such as the original number of items in the collection.

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