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

Problem
Primary curriculum
Secondary curriculum
### Number Lines in Disguise

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

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fractions Rectangle

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triathlon and Fitness

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nutrition and Cycling

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Speed-time Problems at the Olympics

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ratios and Dilutions

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Speeding Boats

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Chance to Win?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Temperature

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All about Ratios

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Equal Temperament

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Trapezium Four

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mixing More Paints

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mixing Paints

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cereal Mix

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Little Man

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Blackcurrantiest

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

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle in a Triangle

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Orange Drink

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pumpkin Pie Problem

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Oh for the Mathematics of Yesteryear

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sitting Pretty

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One and Three

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rule of Three

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Ladders

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Areas and Ratios

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Thoughts

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Ratio

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rati-o

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Racing Odds

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Semi-square

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pent

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.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rhombus in Rectangle

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Around and Back

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Five Circuits, Seven Spins

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Do Unto Caesar

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Big?

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Same Height

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Star Gazing

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### From All Corners

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Pegs

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Burning Down

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gift of Gems

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Contact

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plane to See

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Napoleon's Hat

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### At a Glance

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

Age 14 to 16

Challenge Level