Ratio Problems

A ratio compares two amounts with a colon, like 3:5 for red to blue beads. It does not say there are exactly 3 and 5 beads; it says every 3 reds come with 5 blues. Double the batch and 6:10 means exactly the same mix, because both sides grew by the same multiplier. To find a missing value, find what one part is worth. With 15 blue beads in a 3:5 mix, each part is 15 divided by 5 = 3 beads, so red is 3 parts of 3 = 9 beads. The multiplier (here 3) works on both sides at once, which keeps the mix unchanged.

Ratio tables lay out equivalent ratios row by row so the pattern shows. For 2 apples to 5 oranges, the table reads 2:5, 4:10, 6:15, 8:20. Each step adds one more copy of the base ratio, and the multiplier column (1, 2, 3, 4) tells you how far you scaled. Tables also catch additive traps. Someone who adds 2 to each side of 2:5 gets 4:7, which breaks the table pattern and tastes wrong in a recipe. Only multiplying both sides preserves the ratio, and the table proves it at a glance.

Ratios describe real situations: juice concentrate to water, flour to sugar, wins to matches. The order matters: 3:5 for red:blue is different from 5:3. Always label which number belongs to which thing before calculating, or the answer describes the wrong mix. Scaling a ratio answers how much questions. Twenty oranges at 2 apples per 5 oranges means 4 groups of the base pair, so 4 times 2 = 8 apples. Division finds the group count, multiplication finds the missing amount: that pair solves almost every ratio problem.

A proportion is an equation of two equal ratios, such as 3/5 = 9/15. Cross products test equality fast: 3 times 15 = 45 and 5 times 9 = 45, so the ratios match. Unequal cross products expose a broken ratio immediately. Proportions also solve for unknowns: setting 3/5 = r/15 and cross multiplying gives 5r = 45, so r = 9. The unit rate view agrees: 3 reds per 5 blues is 0.6 reds per blue, and 0.6 times 15 is 9.