You can put this solution on YOUR website! Given the following proportion to solve:
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A "shortcut" way to solve proportion problems such as this on is to start with a method called "cross-multiplication." This involves multiplying the numerator on one side by the denominator of the other side. Then multiplying the numerator on the other side by denominator of the opposite side. Finally set these two products equal.
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In other words, if you have a proportion:
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Multiply the numerator of the left side (A) times the denominator of the right side (D). Then multiply the numerator of the right side (C) times the denominator of the left side (B). Note that if you draw a line from A to D (representing the two terms to be multiplied) and then draw another line connecting C and B (representing the other two terms to be multiplied) these two lines form a cross through the equal sign. That's where the term "cross-multiplication" comes from.
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Let's try this method on the problem you were given.
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Multiply the left side numerator (x + 1) times the right side denominator (x + 2) using the FOIL method (multiply First terms, then Outside terms, then Inside terms, and finally Last terms).
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This multiplication results in:
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Simplify this product by adding together the two terms containing x and you find this product to be:
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Remember this first cross-product. Next do the second cross-product by multiplying the numerator on the right side of the proportion (10) by the denominator on the left side of the proportion (3) and you find the second cross-product to be:
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Now, set your two cross-products equal as follows:
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This is a quadratic equation. Begin solving it by getting it into the standard form of:
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Do that by subtracting 30 from both sides to get:
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Note that this quadratic can be factored as follows:
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This equation will be true if either of the two factors equals zero because a zero multiplier on the left side makes the product equal to the zero on the right side. So either:
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Solving these two equations results in the two possible answers. Either:
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Check by returning to the original proportion and first substituting -7 for x to get:
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which simplifies to:
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and this further simplifies to:
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This shows that when x = -7 both sides of the proportion are equal. So this answer checks out.
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Then again return to the original proportion and substitute +4 for x to get:
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and this results in:
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Note that if you divide both the numerator and the denominator on the right side by 2, the proportion becomes:
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So this answer also works.
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Both answers check out. The two answers to this problem are x = -7 and x = 4.
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Hope this cross-multiplication method helps you with understanding a way to work proportion problems. It works with any proportion that sets two ratios equal. Just cross-multiply, set the two products equal, and solve.
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