Math: Multi-Step Problems That Mix the Strands
The guide's own promise, drawn out: familiar skills braided into complex multi-step items where ratio, percent, units and geometry all land in one question.
13 topics · 48 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Ratio and Perimeter in One Problem
- How it worksHow does letting the sides be 3x and 2x turn a ratio problem into a one-variable equation?
- Worked exampleFind the dimensions of a rectangle with a perimeter of 510 cm whose length to width ratio is 3 to 2.
- Worked exampleFind the sides of a triangle with a perimeter of 96 cm whose sides come in the ratio 3 to 4 to 5.
- Common mistakeWhy doesn't a 3 to 2 ratio mean the rectangle measures 3 cm by 2 cm?
Your turneasy1 of 3
A rectangle has a perimeter of P centimetres and a length to width ratio of 7 to 2. Write an expression for its width, in centimetres, in terms of P.Ratio and Area in One Problem
- How it worksHow does a side ratio of 2 to 3 grow into an area ratio of 4 to 9?
- Worked exampleFind the area of a rectangle whose sides are in the ratio 5 to 3 and whose perimeter is 128 m.
- Going furtherWhat happens to the ratio of two volumes when the ratio of matching edges is 2 to 5?
Your turnhard1 of 3
A rectangle has sides in the ratio 7 to 3 and a perimeter of P metres. Write an expression for its area, in square metres, in terms of P.Unit Conversion Inside a Geometry Problem
- How it worksHow do you convert a length before an area calculation rather than patching it up afterwards?
- Worked exampleFind the area in square feet of a rectangle measuring 30 inches by 4 feet.
- Worked exampleFind the gallons filling a tank of 4 by 3 by 2 feet when one cubic foot holds 7.48 gallons.
- Common mistakeWhy isn't one square foot equal to 12 square inches?
Your turneasy1 of 3
Find the area, in square feet, of a rectangle measuring 30 inches by 9 feet.Chained Conversions and Rate Problems
- How it worksHow do you chain conversion factors so the unwanted units cancel on the way through?
- Worked exampleFind the dollars for 140 lorgs and 16 dalts when a dollar buys 7 lorgs and a dollar buys 0.5 dalt.
- Worked exampleFind the minutes a 45 km trip takes at 60 km per hour, then the minutes saved at 90 km per hour.
Your turneasy1 of 3
Find the number of minutes a 93 kilometre trip takes at 60 kilometres per hour, minus the number it takes at 90 kilometres per hour.Percent Applied to an Area
- How it worksHow does a percent change on each dimension multiply into a percent change on the area?
- Worked exampleFind the percent change in area when a rectangle's length grows 20 percent and its width drops 10 percent.
- Worked exampleFind the lawn left over when a 30 by 20 m yard gives up 35 percent of its area to a patio.
- Common mistakeWhy doesn't a 25 percent rise followed by a 25 percent fall bring a number back where it started?
Your turnmedium1 of 3
A rectangle's length grows by p percent while its width drops by 22 percent. Write an expression, in terms of p, for the new area as a percentage of the old one.Turning a Geometric Condition into an Equation
- How it worksHow do you name one unknown and then write every other length in terms of that one letter?
- How it worksHow do you spot which quantity in a geometry word problem deserves to become x?
- Worked exampleFind the width of a rectangle whose length is 5 cm more than twice the width and whose perimeter is 64 cm.
- Worked exampleFind a triangle's angles when the second is twice the first and the third is 20 degrees above the first.
- Common mistakeWhy doesn't picking the wrong quantity as x make a problem unsolvable rather than merely slower?
Your turneasy1 of 3
A rectangle's length is 8 centimetres more than twice its width, and its perimeter is P centimetres. Write an expression for the width, in centimetres, in terms of P.Fractions and Decimals Inside Geometry
- How it worksHow do you multiply three fractional edge lengths without converting anything into a decimal?
- Worked exampleFind the volume of a box measuring two and a half by one and a third by 3 feet.
- Worked exampleFind the area of a triangle with a base of 4.5 cm and a height of 6.2 cm without a calculator.
Your turnmedium1 of 3
Find the volume, in cubic feet, of a box measuring two and a half feet by one and a third feet by 7 feet.Statistics Folded Into a Word Problem
- How it worksHow do you spot whether a word problem wants a mean, a median or just a straight sum?
- Worked exampleFind the average speed of a car covering 120 km at 60 km per hour and 120 km at 40 km per hour.
- Worked exampleFind the new class average when a 30-student class averaging 76 gains a student who scores 96.
- Common mistakeWhy isn't the middle of 60 and 40 km per hour the right average speed over two equal distances?
Your turnhard1 of 3
A car covers 120 kilometres at 60 kilometres per hour and another 120 kilometres at 72 kilometres per hour. Find the average speed for the whole journey, in kilometres per hour.Probability on a Geometric Figure
- How it worksHow does the ratio of a shaded area to the whole area become the chance of a random hit?
- Worked exampleFind the chance a random point in a 10 by 10 square lands inside the circle of radius 5 inside it.
- Going furtherWhat happens to that chance when the circle is swapped for a square of half the outer side?
Your turnmedium1 of 3
A circle of radius 5 sits inside a rectangle measuring 24 by 10. Find the probability that a random point of the rectangle lands inside the circle.Working Multi-Step Problems Without a Calculator
- How it worksHow does holding answers as fractions and as π until the last step save you from ugly decimals?
- Worked exampleFind 15 percent of 340 and then two thirds of that result using mental arithmetic only.
- Worked exampleFind the area of a circle of radius 14 using 22 over 7, then double it, all by hand.
- Going furtherWhat happens to your accuracy when you round early inside a multi-step geometry problem?
Your turneasy1 of 3
Find 15 percent of 687, and then two thirds of that result, using mental arithmetic only.Checking Your Answer Against the Choices
- How it worksHow do you rewrite an answer into an equivalent form when it seems to be missing from the options?
- Worked exampleFind which of 150 by 105, 153 by 102, 158 by 97 and 165 by 90 fits a perimeter of 510 at a 3 to 2 ratio.
- Common mistakeWhy isn't an answer wrong just because it fails to match any option on the first look?
Your turneasy1 of 3
A rectangle has a perimeter of 609 centimetres and a length to width ratio of 3 to 2. Find the length, in centimetres.Drawing the Figure Yourself
- How it worksHow do you label a sketch so that every number in the question lands somewhere on it?
- Worked exampleFind the area of a path 2 m wide running around the outside of a 12 by 8 m rectangular garden.
- Worked exampleFind the straight-line gap between two points 9 km north and 12 km east of one corner by sketching first.
- Going furtherWhat happens when a sketch comes out badly out of proportion but every label on it is correct?
Your turnhard1 of 3
A path 3 metres wide runs around the outside of a rectangular garden measuring 16 metres by w metres. Write an expression, in terms of w, for the area of the path, in square metres.Putting Every Strand Into One Question
- How it worksHow do you break an item mixing fractions, equations and averages into steps you can order?
- Worked exampleFind the cost of tiling a 15 by 12 ft floor at 3.50 dollars a square foot with 8 percent tax on top.
- Worked exampleFind the radius of a cylinder holding 1540 cubic cm at a height of 10 cm, using 22 over 7 for π.
- Ties togetherHow do the geometry rules, the percent rules and the averages all meet inside one multi-step item?
Your turnmedium1 of 3
Write an expression, in terms of w, for the cost of tiling a floor measuring 18 feet by w feet at 3.8 dollars a square foot, with 10 percent tax added on top.