There are 102 NRICH Mathematical resources connected to Ratio and proportion, you may find related items under Fractions, decimals, percentages, ratio and proportion.
Broad Topics > Fractions, decimals, percentages, ratio and proportion > Ratio and proportion
Number Lines in Disguise
Age 7 to 14
Challenge Level
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Fractions Rectangle
Age 11 to 14
Challenge Level
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?
Triathlon and Fitness
Age 11 to 14
Challenge Level
The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories?
Nutrition and Cycling
Age 14 to 16
Challenge Level
Andy wants to cycle from Land's End to John o'Groats. Will he be able to eat enough to keep him going?
Speed-time Problems at the Olympics
Age 14 to 16
Challenge Level
Have you ever wondered what it would be like to race against Usain Bolt?
Ratios and Dilutions
Age 14 to 16
Challenge Level
Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions
Speeding Boats
Age 14 to 16
Challenge Level
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
A Chance to Win?
Age 11 to 14
Challenge Level
Imagine you were given the chance to win some money... and imagine you had nothing to lose...
Temperature
Age 11 to 14
Challenge Level
Is there a temperature at which Celsius and Fahrenheit readings are the same?
All about Ratios
Age 16 to 18
Challenge Level
A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.
Equal Temperament
Age 14 to 16
Challenge Level
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.
Trapezium Four
Age 14 to 16
Challenge Level
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Mixing More Paints
Age 14 to 16
Challenge Level
Can you find an efficent way to mix paints in any ratio?
Mixing Paints
Age 11 to 14
Challenge Level
Can you work out how to produce different shades of pink paint?
Cereal Mix
Age 11 to 14
Challenge Level
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?
Little Man
Age 5 to 7
Challenge Level
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?
Blackcurrantiest
Age 5 to 7
Challenge Level
Can you decide whose drink has the strongest blackcurrant flavour from these pictures?
Triangle in a Triangle
Age 14 to 16
Challenge Level
Can you work out the fraction of the original triangle that is covered by the inner triangle?
Orange Drink
Age 7 to 11
Challenge Level
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?
Pumpkin Pie Problem
Age 7 to 11
Challenge Level
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?
Oh for the Mathematics of Yesteryear
Age 11 to 14
Challenge Level
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?
Sitting Pretty
Age 14 to 16
Challenge Level
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?
One and Three
Age 14 to 16
Challenge Level
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?
Rule of Three
Age 11 to 14
Challenge Level
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
Two Ladders
Age 14 to 16
Challenge Level
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?
Areas and Ratios
Age 16 to 18
Challenge Level
Do you have enough information to work out the area of the shaded quadrilateral?
Golden Thoughts
Age 14 to 16
Challenge Level
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.
Golden Ratio
Age 16 to 18
Challenge Level
Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.
Rati-o
Age 11 to 14
Challenge Level
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?
Racing Odds
Age 11 to 14
Challenge Level
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?
Semi-square
Age 14 to 16
Challenge Level
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?
Pent
Age 14 to 18
Challenge Level
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.
Rhombus in Rectangle
Age 14 to 16
Challenge Level
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.
Around and Back
Age 14 to 16
Challenge Level
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.
Five Circuits, Seven Spins
Age 16 to 18
Challenge Level
A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.
Do Unto Caesar
Age 11 to 14
Challenge Level
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?
How Big?
Age 11 to 14
Challenge Level
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Same Height
Age 14 to 16
Challenge Level
A trapezium is divided into four triangles by its diagonals. Can you work out the area of the trapezium?
Star Gazing
Age 14 to 16
Challenge Level
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
From All Corners
Age 14 to 16
Challenge Level
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.
Square Pegs
Age 11 to 14
Challenge Level
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Burning Down
Age 14 to 16
Challenge Level
One night two candles were lit. Can you work out how long each candle was originally?
Gift of Gems
Age 14 to 16
Challenge Level
Four jewellers share their stock. Can you work out the relative values of their gems?
Contact
Age 14 to 16
Challenge Level
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?
Plane to See
Age 16 to 18
Challenge Level
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.
Napoleon's Hat
Age 16 to 18
Challenge Level
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?
At a Glance
Age 14 to 16
Challenge Level
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
NRICH is part of the family of activities in the Millennium Mathematics Project.