Chapter 10.4 — Exercise 10.4 — Compound Proportion

Three Ways the Combination Can Work

Depending on how the main quantity relates to each of the other two, a compound proportion takes one of three forms:

Each of these three formulas is really just the ordinary two-quantity proportion rule from the last two exercises, applied twice in a row and then combined — nothing about compound proportion needs a genuinely new idea, only careful bookkeeping about which quantities pull in which direction.

All Direct: More People, More Days, More Cost

Rice costing ₹480 feeds 8 members for 20 days. Both more members and more days push the cost up, so cost is directly proportional to both, and the first formula from the table applies:

Feeding more people for more days naturally costs more than either change would on its own — the compound formula captures both increases in one calculation rather than requiring two separate proportion problems chained together.

Mixed Case: Direct With Length, Inverse With Days

10 men lay a 75 km road in 5 days. How many days do 15 men need to lay a 45 km road? Here men are directly proportional to the length of road (more road needs more workers) but inversely proportional to the number of days (more workers finish faster) — the third pattern from the table:

A second mixed-case problem follows an identical shape: 175 men dig a 3150 m canal in 36 days; how many men are needed for a 3900 m canal in 24 days? Men are again directly tied to canal length and inversely tied to the number of days available:

Both problems combine a growing quantity (more kilometres of road or canal) with a shrinking one (fewer days allowed) pulling against each other on the same side of the equation — exactly what the z₁/y₁ = z₂/y₂-style term in the formula is built to handle.

All Inverse: Fewer Hours, Fewer Days, But More Workers Needed

24 men working 8 hours a day finish a job in 15 days. How many days do 20 men working 9 hours a day need for the same job? Fewer men working the same total job takes longer, and more hours per day finishes it sooner — both hours and days move inversely against the number of men, matching the second formula:

A second all-inverse problem: 14 typists working 6 hours a day take 12 days to finish a manuscript. How long would 4 typists working 7 hours a day take?

Dropping from 14 typists to just 4 is a big cut in workforce, only partly offset by the extra hour worked each day — which is exactly why the job stretches from 12 days out to 36, a threefold increase, rather than something more modest. It's worth checking the two effects separately to see why 36 is the right order of magnitude: cutting the typist count from 14 to 4 alone (holding hours fixed) would stretch 12 days out to 12 × 14/4 = 42 days, and the extra hour per day then shortens that slightly, from 42 days down to 42 × 6/7 = 36 days — the same final answer, reached by applying the two proportional effects one after another instead of in a single combined step.

Reading the Three Formulas as One Idea

It's worth seeing why the three formulas in the opening table aren't three separate rules to memorise. In every case, x is compared across two situations by comparing y and z across those same two situations — the only choice being made is whether each of y and z sits on the "same side" as x (direct) or the "opposite side" (inverse). Rice cost sits on the same side as both members and days, giving the all-direct formula. Typists sit opposite both hours and days, giving the all-inverse formula. Road-laying men sit with length but against days, giving the mixed formula. Once that same-side-or-opposite-side judgment is made correctly for each of the two secondary quantities, the formula itself follows automatically — there's no need to memorise three formulas as unrelated facts.

What Goes Wrong If a Relationship Is Assumed Backwards

The single most common slip in compound proportion isn't arithmetic — it's misjudging whether a quantity belongs on the direct or inverse side of the formula before any calculation starts. Take the road-building problem again: 10 men lay 75 km in 5 days. It's tempting to assume men and days are directly proportional (more men, more days), but that's backwards — more men working the same stretch of road finish it sooner, not later, which is exactly why days sits inversely against the number of men in the correct formula. Running the calculation with the relationship flipped the wrong way doesn't produce an error message or an obviously impossible number — it produces a wrong answer that looks perfectly reasonable, which is precisely what makes this mistake dangerous. The safeguard is always the same one used throughout this chapter: before writing any ratio, ask in plain words whether increasing one quantity should increase or decrease the other, and only then decide which of the three formulas applies.

The Arc of This Chapter, in One Line

This chapter moved from two quantities changing together (Exercise 10.1), to two quantities changing in opposite directions (10.2 and 10.3), to three quantities linked together at once here in Exercise 10.4 — the same core ratio idea, applied to steadily more realistic situations. The next chapter, Algebraic Expressions, moves away from ratios entirely and into building and simplifying expressions with variables — though the (x+1)-style algebra from Exercise 10.3's general proportion problem is a small preview of what's ahead.