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Resources tagged with Ratio similar to Golden Thoughts:

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Broad Topics > Fractions, Decimals, Percentages, Ratio and Proportion > Ratio

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Golden Thoughts

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

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Pent

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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Equal Temperament

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

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Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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Sitting Pretty

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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Ratio Pairs 3

Stage: 3 and 4 Challenge Level: Challenge Level:1

Match pairs of cards so that they have equivalent ratios.

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Six Notes All Nice Ratios

Stage: 4 Challenge Level: Challenge Level:1

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

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How Big?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

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One and Three

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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Ratio or Proportion?

Stage: 2 and 3

An article for teachers which discusses the differences between ratio and proportion, and invites readers to contribute their own thoughts.

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Ratio Sudoku 2

Stage: 3 and 4 Challenge Level: Challenge Level:1

A Sudoku with clues as ratios.

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Do Unto Caesar

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the. . . .

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Mixing More Paints

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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Mixing Paints

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

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Speeding Boats

Stage: 4 Challenge Level: Challenge Level:1

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

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Triangle in a Triangle

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you work out the fraction of the original triangle that is covered by the inner triangle?

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Trapezium Four

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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Cereal Mix

Stage: 3 Challenge Level: Challenge Level:1

A farmer is supplying a mix of seeds, nuts and dried apricots to a manufacturer of crunchy cereal bars. What combination of ingredients costing £5 per kg could he supply?

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Pythagoras’ Comma

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

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From All Corners

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

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Ratio Sudoku 1

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

A Sudoku with clues as ratios.

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Semi-square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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Rhombus in Rectangle

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

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Around and Back

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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A Scale for the Solar System

Stage: 4 Challenge Level: Challenge Level:1

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

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Racing Odds

Stage: 3 Challenge Level: Challenge Level:1

In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?

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Ratio Sudoku 3

Stage: 3 and 4 Challenge Level: Challenge Level:1

A Sudoku with clues as ratios or fractions.

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Star Gazing

Stage: 4 Challenge Level: Challenge Level:1

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

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Exact Dilutions

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Which exact dilution ratios can you make using only 2 dilutions?

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The Rescaled Map

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

We use statistics to give ourselves an informed view on a subject of interest. This problem explores how to scale countries on a map to represent characteristics other than land area.

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Mixing Lemonade

Stage: 3 Challenge Level: Challenge Level:1

Can you work out which drink has the stronger flavour?

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Ratios and Dilutions

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions

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Oh for the Mathematics of Yesteryear

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45. . . .

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Rati-o

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

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Tin Tight

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

What's the most efficient proportion for a 1 litre tin of paint?

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Bus Stop

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Two buses leave at the same time from two towns Shipton and Veston on the same long road, travelling towards each other. At each mile along the road are milestones. The buses' speeds are constant. . . .

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Circuit Training

Stage: 4 Challenge Level: Challenge Level:1

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

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Points in Pairs

Stage: 4 Challenge Level: Challenge Level:1

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

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Slippage

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum. . . .

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Pi, a Very Special Number

Stage: 2 and 3

Read all about the number pi and the mathematicians who have tried to find out its value as accurately as possible.