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Original
2026-01-01
Modified
2026-02-28
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<p>Proportions follow specific properties that make it easier to solve ratio-related problems. The properties of proportions are: </p>
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<p>Proportions follow specific properties that make it easier to solve ratio-related problems. The properties of proportions are: </p>
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<ul><li>Cross Multiplication Property</li>
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<ul><li>Cross Multiplication Property</li>
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<li>Invertendo Property</li>
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<li>Invertendo Property</li>
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<li>Alternendo Property</li>
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<li>Alternendo Property</li>
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<li>Componendo and Dividendo Property</li>
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<li>Componendo and Dividendo Property</li>
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<li>Mean Proportional Property</li>
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<li>Mean Proportional Property</li>
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</ul><h3><strong>Cross Multiplication Property:</strong></h3>
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</ul><h3><strong>Cross Multiplication Property:</strong></h3>
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<p>In a proportion a/b = c/d, the<a>product</a>of the means equals the product of the extremes: \(a × d = b × c\). </p>
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<p>In a proportion a/b = c/d, the<a>product</a>of the means equals the product of the extremes: \(a × d = b × c\). </p>
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<p><strong>For example,</strong> If \({4 \over 6} = {6 \over 9} \), then \(4 \times 9 = 6 \times 6 = 36\)</p>
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<p><strong>For example,</strong> If \({4 \over 6} = {6 \over 9} \), then \(4 \times 9 = 6 \times 6 = 36\)</p>
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<h3><strong>Invertendo Property:</strong></h3>
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<h3><strong>Invertendo Property:</strong></h3>
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<p>Invertendo property states that if two ratios are equal, then their reciprocal is also equal. If \({{a \over b} = {c \over d}} \implies {{b \over a} = {d \over c}} \)</p>
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<p>Invertendo property states that if two ratios are equal, then their reciprocal is also equal. If \({{a \over b} = {c \over d}} \implies {{b \over a} = {d \over c}} \)</p>
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<p>For example, \({8 \over 10} = {16 \over 20} \implies {10\over 8} = {20\over 16}\)</p>
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<p>For example, \({8 \over 10} = {16 \over 20} \implies {10\over 8} = {20\over 16}\)</p>
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<h3><strong>Componendo and Dividendo Property:</strong></h3>
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<h3><strong>Componendo and Dividendo Property:</strong></h3>
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<p>According to the componendo property, when we add the<a>numerator and denominator</a>of each ratio to forms a new ratio. If \({a \over b} = {c \over d}\) then \({{a + b} \over b} = {{c + d} \over d}\). </p>
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<p>According to the componendo property, when we add the<a>numerator and denominator</a>of each ratio to forms a new ratio. If \({a \over b} = {c \over d}\) then \({{a + b} \over b} = {{c + d} \over d}\). </p>
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<p> </p>
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<p> </p>
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<p> For example, if \({5 \over 7} = {15\over 21}\), then \({{5 + 7} \over 7} = {{15 + 21} \over 21} \implies {12 \over 7} = {36 \over 21}\)</p>
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<p> For example, if \({5 \over 7} = {15\over 21}\), then \({{5 + 7} \over 7} = {{15 + 21} \over 21} \implies {12 \over 7} = {36 \over 21}\)</p>
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<p>Dividendo property states that, subtract the denominator from the numerator of each ratio to form a new ratio. If \({a \over b} = {c \over d} \implies {{a - b} \over b} = {{c - d} \over d}\)</p>
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<p>Dividendo property states that, subtract the denominator from the numerator of each ratio to form a new ratio. If \({a \over b} = {c \over d} \implies {{a - b} \over b} = {{c - d} \over d}\)</p>
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<p>For example, if \({5 \over 7} = {15\over 21}\), then \({{5 - 7} \over 7} = {{15 - 21} \over 21} \implies {-2 \over 7} = {-6 \over 21}\)</p>
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<p>For example, if \({5 \over 7} = {15\over 21}\), then \({{5 - 7} \over 7} = {{15 - 21} \over 21} \implies {-2 \over 7} = {-6 \over 21}\)</p>
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<h3><strong>Mean Proportional Property:</strong></h3>
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<h3><strong>Mean Proportional Property:</strong></h3>
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<p>The<a>mean</a>proportional property states that, if there are three quantities a, b, and c are in continued proportion, then b is the mean proportional between a and c. </p>
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<p>The<a>mean</a>proportional property states that, if there are three quantities a, b, and c are in continued proportion, then b is the mean proportional between a and c. </p>
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<p>If \({a \over b} = {b \over c} \implies b^2 = a \times c\) </p>
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<p>If \({a \over b} = {b \over c} \implies b^2 = a \times c\) </p>
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<p>For example, if 4, \(x\), and 9 are in continued proportion, then: </p>
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<p>For example, if 4, \(x\), and 9 are in continued proportion, then: </p>
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<p>\({4 \over x} = {x \over 9}\)</p>
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<p>\({4 \over x} = {x \over 9}\)</p>
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<p>\(x^2 = 4 \times 9 = 36\)</p>
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<p>\(x^2 = 4 \times 9 = 36\)</p>
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<p>\(x\) \(=\) \(6\) </p>
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<p>\(x\) \(=\) \(6\) </p>
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