# Equations - August 2007, All Stages

## Problems

### Number Lines

##### Stage: 1 Challenge Level:

Leah and Tom each have a number line. Can you work out where their counters will land? What are the secret jumps they make with their counters?

### Secret Number

##### Stage: 1 Challenge Level:

Annie and Ben are playing a game with a calculator. What was Annie's secret number?

### Getting the Balance

##### Stage: 1 Challenge Level:

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

### Shape Times Shape

##### Stage: 2 Challenge Level:

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

### Balance of Halves

##### Stage: 2 Challenge Level:

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

### Two Number Lines

##### Stage: 2 Challenge Level:

Max and Mandy put their number lines together to make a graph. How far had each of them moved along and up from 0 to get the counter to the place marked?

### Parallel Lines

##### Stage: 3 Challenge Level:

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

### Clocking Off

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

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

### Temperature

##### Stage: 3 Challenge Level:

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

### Matchless

##### Stage: 4 Challenge Level:

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

### Gutter

##### Stage: 4 Challenge Level:

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

### Perpendicular Lines

##### Stage: 4 Challenge Level:

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

### Reasonable Algebra

##### Stage: 4 Challenge Level:

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

### Equation Attack

##### Stage: 5 Challenge Level:

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

### Production Equation

##### Stage: 5 Challenge Level:

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

### Resistance

##### Stage: 5 Challenge Level:

Find the equation from which to calculate the resistance of an infinite network of resistances.