Fractions in ratio problems: split and compare
Number • Fractions, decimals and percentages
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Key concepts
What you'll likely be quizzed about
Definition and connection between ratios and fractions
A ratio a:b compares the sizes of two parts. The total number of parts equals a + b. Each part of the ratio becomes a fraction of the whole by placing the part value over the sum of the parts. Therefore a becomes the fraction a/(a+b) and b becomes b/(a+b). The fractions always sum to 1 because the numerator sum equals the denominator. Conversion from ratio to fraction simplifies comparison and calculation. A ratio expressed as fractions allows multiplication by a known total to find actual quantities. When the ratio contains more than two parts, the same principle applies: each part over the sum of all parts gives the fraction for that part.
Splitting a quantity using fractions from a ratio
When the total quantity is known, multiplication of the total by each fraction gives the size of each part. Cause: the fraction represents the portion of the whole. Effect: multiplying the total by that fraction yields the part value. Example: For a ratio 2:3 and total 50, the parts are 2/(2+3)×50 and 3/(2+3)×50, producing 20 and 30. Limiting factors include non-integer results and units. When results require whole-number answers, rounding rules or reinterpretation of the problem may apply. Exact fractional answers remain valid when units allow fractions (e.g., litres, kilograms).
Scaling ratios and equivalent fractions
Ratios remain equivalent when both terms multiply by the same factor. Cause: multiplication of both parts preserves the relative sizes. Effect: the corresponding fractions remain equal because both numerator and denominator scale by the same factor. Example: 2:3 equals 4:6, and 2/5 equals 4/10. Scaling helps when matching ratios to known totals or when comparing ratios with different total parts. When converting from fractions back to a ratio, multiply fractions by a common denominator to obtain integer parts if integer ratios are required.
Comparing parts and checking results
Comparison of parts uses fractions to show relative sizes directly because fractions share the same denominator (the total parts). Cause: expressing parts as fractions places each part over the same whole. Effect: direct comparison of numerators indicates larger or smaller shares. Example: For ratios 3:5 and 4:7, convert each to decimal or compare fractions 3/8 vs 4/11 to determine the larger share. Verification uses addition of fractional parts to check that they sum to 1 or that scaled parts sum to the known total. Mismatch signals calculation error or incorrect interpretation of the ratio.
Key notes
Important points to keep in mind