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There are **21** NRICH Mathematical resources connected to **Other equations**, you may find related items under Algebraic expressions, equations and formulae.

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
### Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Polynomial Relations

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### N Is a Number

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?

Age 11 to 14

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
### Top-heavy Pyramids

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Are You Kidding

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Deep Roots

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

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
### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Four Five

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hike and Hitch

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the road (at 4 miles per hour)...

Age 14 to 16

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
### Hand Swap

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Not a Polite Question

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

Age 11 to 14

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Coffee

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Our Ages

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Age 16 to 18

Challenge Level