Directly proportional

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Whenever one ratio and one of the others is known, the unknown can be solved. For two quantities with the same proportion, we say that one (y) is proportional to the second (x) and it is written symbolically as

where k is the constant of proportionality. The ratio can be expressed as a line through the origin.

The more of x the more of y.

It can be also thought as an mathematical equation, where two ratios of two quantities are equal:

If any three of these parameters are known, the missing value can be solved with cross-multiplication as

Example 1

The map scale is 1:15000. If the distance of the two houses in the map is about 5 cm, what is the distance in reality? The ratio of the real measure and the map measure has to be the same 1:15000 all the time. That is

We know the map measure to be 5 cm = 50 mm and measures are usually given in millimetres, so

Example 2.

A restaurant-keeper bought 4.2 kilos of strawberries with 16.50 euros. Amount of 380 g of these strawberries were used for a cake. How much should be marked as a product cost for strawberries of the cake? The proportion of the price and amount of strawberries must be the same. Thus,

Example 3.

A recipe for 6 persons requires 500 grams of minced meat. There were 23 persons coming to the party, so how much minced meat should be bought at least?

Example 4.

How much is 28 590 Japanese jeni in euros, if one euro is 122.04 JPY? Change reward is not paid attention.

Example 5.

Linda and Ann both needed some extra wine glasses at the same time. They noticed, that they can get them cheaper, if they rent a bigger amount of them and divide them. Linda needed 250 extra wine glasses and Ann 130. The total price for the rent 165 euros including all extra costs and value-added tax. If they share the total price in the same ratio as they needed glasses, how much should each of them to pay? Let us solve first the cost for Linda. As Linda needed 250 from total of 380 glasses, we know the ratio and we can form the proportion:

. Thus, Linda pays 108.55 euros and Ann has to pay 165 euros - 108.55 euros = 56.45 euros.

Example 6.

The area of an apartment and the maintanance charge of the apartment are direct variation: the bigger the apartment, the more you have to pay. As the below applet shows us, the graph expressing the direct variation is the straight line that goes through the origin. In the graph A is the area of the apartment in m² and B is the monthly maintanance charge in euros.