There are **102** NRICH Mathematical resources connected to **Ratio and proportion**, you may find related items under Fractions, decimals, percentages, ratio and proportion.

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Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

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The large rectangle is divided into a series of smaller quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

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The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?

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Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?

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Have you ever wondered what it would be like to race against Usain Bolt?

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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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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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Imagine you were given the chance to win some money... and imagine you had nothing to lose...

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Is there a temperature at which Celsius and Fahrenheit readings are the same?

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A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

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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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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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Can you find an efficent way to mix paints in any ratio?

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Can you work out how to produce different shades of pink paint?

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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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The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

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Can you decide whose drink has the strongest blackcurrant flavour from these pictures?

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

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A 750 ml bottle of concentrated orange squash is enough to make fifteen 250 ml glasses of diluted orange drink. How much water is needed to make 10 litres of this drink?

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Peter wanted to make two pies for a party. His mother had a recipe for him to use. However, she always made 80 pies at a time. Did Peter have enough ingredients to make two pumpkin pies?

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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 days against the siege of the enemy?

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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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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 metres from B. How long is the lake?

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If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second wall. At what height do the ladders cross?

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Do you have enough information to work out the area of the shaded quadrilateral?

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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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Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

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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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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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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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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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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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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 around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.

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A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

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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 assets of the players at the beginning of the evening?

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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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A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?

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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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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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Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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One night two candles were lit. Can you work out how long each candle was originally?

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Four jewellers share their stock. Can you work out the relative values of their gems?

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

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Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?