Word Problem Setup Helper: Catch-Up, Work Together, Mixtures, Ratios

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Opens on Chapter 21’s example: Otto’s van at 50 mph, and a second van leaving an hour later at 60 mph.

Problem

First speed

Second speed (leaves later)

Head start (hours)

Enter your numbers, then press Work it out.

How it works

Each family has its own picture. Catching up: when the second catches the first, both have gone the same distance, so speed × time is equal: 50t = 60(t − 1). Working together: rates add, so a job taking 3 hours alone is 1/3 of the job an hour; together T solves T/3 + T/6 = 1 (never the average of the times). Dilution: water adds volume but no concentrate, so the concentrate stays the same. Ratio: 3 : 5 is 8 equal parts. Percent backward: divide by the multiplier, never take the percent off the new figure. For whole items, always round up.

From the book, Chapter 21. The second van catches up 6 hours after Otto set off (5 hours after it left), 300 miles out. Roz and Teo paint the room in 2 hours together (not 4.5). 10 gallons of a 30% mix hold 3 gallons of concentrate; 5 gallons of water make 15 gallons at 20%. $1,200 in the ratio 3 : 5 is $150 a part: $450 and $750. $54 after a 20% rise was $45 (not $43.20). Three seats adding to 87 start at 28; 130 people need 2.71, so 3 buses. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.