The Ratio That Runs Backwards Proportional Reasoning-2

Before anything new, this chapter opens by revisiting a familiar idea: two quantities are in a proportional relationship whenever they change together by the same factor, written as a ratio like 2:1. A homely example makes the point concrete -- idli batter is traditionally mixed from rice and urad dal, and two cooks, Viswanath and Puneet, use different-looking quantities: 6 cups of rice with 3 cups of urad dal, versus 4 cups of rice with 2 cups of urad dal. Do their batters taste the same? Testing this with cross-multiplication settles it: comparing 6:3 against 4:2 gives 6 x 2=12 and 3 x 4=12 -- equal products confirm the two ratios are genuinely proportional, so cooked the same way, the idlis should taste identical. This gives the chapter's first working rule: two ratios a:b and c:d are proportional exactly when a x d equals b x c, or equivalently when a/c equals b/d.

Ratios don't only compare batter or paint -- they quietly sit in the corner of nearly every printed map too, usually written as something like RF = 1:60,00,000. This is a Representative Fraction: it states that one unit of distance measured on the map corresponds to that many of the same units in real life on the ground, so 1 cm on such a map really means 60,00,000 cm -- a tidy 60 km -- between two points in reality. Interestingly, the very map used to introduce this idea, showing cities like Bengaluru, Chennai, and Mangaluru across South India, carries a small printed disclaimer of its own: 'Map not to scale.' That's a genuinely useful reminder hiding in plain sight -- a ratio can be written confidently on a diagram without the diagram actually being drawn to that ratio, which is exactly why the chapter's follow-up activity asks students to check real distances with a ruler on an actual atlas map and compare answers with a classmate, rather than trusting one printed number blindly.

Ratios aren't limited to two quantities either. Viswanath's spice mix powder combines coriander seeds, red chillies, toor dal, and fenugreek seeds in the four-term ratio 8:4:2:1. When Puneet wants to recreate the same flavour but only has 2 red chillies on hand -- exactly half of Viswanath's 4 -- every other ingredient must shrink by that same half-factor too, giving 4:2:1:0.5, still proportional to the original (written 8:4:2:1 :: 4:2:1:0.5). The same idea scales up to genuinely practical situations: mixing a shade of purple paint in the ratio Red:Blue:White::2:3:5, starting from 10 litres of white paint (so each 'part' equals 2 litres), yields 4 litres of red and 6 litres of blue, for 20 litres of purple in total. Or, mixing concrete from cement, sand, and gravel in the ratio 1:1.5:3, starting from 3 bags of cement, scales up to 3:4.5:9 -- a full 16.5 bags of concrete mixture.

Once a ratio is set, splitting any actual total according to it follows one reliable recipe: add up all the ratio's terms, divide the whole amount by that sum to find what a single 'part' is worth, then multiply each term by that value. Needing 110 units of concrete in the ratio 1:1.5:3 (cement:sand:gravel, adding to 5.5 parts), for instance, means dividing 110 by 5.5 to get 20, then scaling every term by 20 -- 20 units of cement, 30 of sand, 60 of gravel. The same recipe, written as a general formula, gives each share of a total x divided in the ratio a:b:c as x times a/(a+b+c), x times b/(a+b+c), and so on -- confirmed again by dividing 50 ml of purple paint in the ratio 2:3:5, which yields 10 ml red, 15 ml blue, and 25 ml white. The same logic extends neatly beyond mixtures, into constructing a triangle whose three angles sit in the ratio 1:3:5: since a triangle's angles always total 180 degrees, dividing 180 by the ratio's 9 parts gives 20 degrees per part, and angles of 20, 60, and 100 degrees.

A pie chart is really just this same ratio-splitting idea, wrapped around a circle instead of a straight line -- because a full circle always measures exactly 360 degrees, and every category's slice should take up a share of that 360 degrees matching its share of the total count. A class of 40 students scoring grades A through E in the counts 12, 10, 8, 6, 4 first simplifies (dividing every term by their HCF of 2) to the ratio 6:5:4:3:2, adding up to 20 parts; 360 divided by 20 gives 18 degrees per part, so Grade A's slice measures 6 x 18=108 degrees, Grade B's 90, Grade C's 72, Grade D's 54, and Grade E's 36 -- a full circle built one measured slice at a time, using nothing more exotic than a protractor and the same 'total divided into parts' reasoning as splitting a bag of coins.

The chapter's real turn comes with a deliberately tempting trap. Puneeth's father rides a motorcycle from Lucknow to Kanpur at 30 km/h, taking 3 hours; switching to a car at 60 km/h, how long will the same trip take? It's tempting to set this up exactly like every proportion so far -- 30:60::3:x -- and solve it as if it were a normal direct proportion. But a table comparing walking (5 km/h, 18 hours), cycling (15 km/h, 6 hours), motorcycle (30 km/h, 3 hours), and car (60 km/h, 1.5 hours) reveals the truth: as speed climbs, travel time keeps shrinking, not growing, and cycling's exact 3-times speed boost over walking is matched by an exact 3-times shrink in travel time (18/6=3). The two quantities change together, by the same factor, but in opposite directions -- a relationship the chapter names inverse proportion, captured by one compact equation: x times y always equals the same constant k (here, the fixed 90 km distance between the two cities), for any pair of matching speed-and-time values.

Once spotted, the same 'x times y=k' pattern shows up in problem after problem. Twenty workers laying a road in 4 days become, if the workforce is halved to 10, a job that takes exactly double the time -- 8 days -- since 20 x 4 must equal 10 times the new number of days. Two pumps filling a tank in 18 hours, with two more pumps added (four pumps total, double the original count), finish in exactly half the time -- 9 hours, since 2 x 18=4 x 9. A school with food for 80 students lasting 15 days, if 20 more students join (100 total), sees its supply drop to just 12 days, since 80 x 15=100 x 12. Each case follows the identical shortcut: whatever factor one quantity is multiplied by, the paired quantity gets divided by that exact same factor, because their product is the one thing that never changes.

Not every combined-effort problem is inverse, though, and the chapter's final worked example makes the contrast explicit. Ram alone cuts a batch of vegetables in 1 hour (so in 1 hour he completes 1 whole 'unit' of that job); Shyam alone takes 1.5 hours (completing 1 divided by 1.5, or two-thirds, of the job in that same hour). Working side by side, their hourly rates simply add: 1 plus two-thirds is five-thirds of the job finished in a single hour. Now the question flips: since five-thirds of a job takes exactly 1 hour, how long does just 1 whole job take? Here, the amount of work done and the time taken to do it are directly proportional -- more time spent together means proportionally more of the job finished -- so five-thirds:1::1:x solves to x=3/5 of an hour, or 36 minutes, for the two of them working together.