# Equations and Formulae - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Equations and Formulae  Age 11-14 and Age 14-16.

Printable worksheets containing selections of these problems are available here.

### Luis' Eight

##### Age 11 to 14 Short Challenge Level:

Luis writes down eight consecutive positive integers. The sum of the three smallest numbers is 33. What is the sum of the three largest numbers?

### Shape Sums

##### Age 11 to 14 Short Challenge Level:

Given some relationships amongst these shapes, how many triangles equal one diamond?

### Seven Dwarfs

##### Age 11 to 14 Short Challenge Level:

The Seven Dwarfs have seven different birthdays. How old can they be?

### Five-to

##### Age 11 to 14 Short Challenge Level:

The number 2005 is the sum of a sequence of five consecutive positive integers. What is the smallest of these integers?

### Partial Magic

##### Age 11 to 14 Short Challenge Level:

This magic square has only been partially completed. Can you still solve it...

### Interstellar

##### Age 11 to 14 Short Challenge Level:

What number could replace * so that */5 is between 3 and 4?

### Fifty Coins

##### Age 11 to 14 Short Challenge Level:

Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?

### Symbol

##### Age 11 to 14 Short Challenge Level:

Using the new operator $\oplus$, can you solve this equation?

### Turnips

##### Age 11 to 14 Short Challenge Level:

Baldrick could buy 6 parsnips and 7 turnips, or 8 parsnips and 4 turnips. How many parsnips could he buy?

### Crossover

##### Age 11 to 14 Short Challenge Level:

The numbers 2 to 8 are to be placed in the diagram so that the row and column each adds to 21. Which numbers can replace x?

### Star Sum

##### Age 11 to 14 Short Challenge Level:

You may have seen magic squares before, but can you work out the missing numbers on this magic star?

### Multiplication Mistake

##### Age 11 to 14 Short Challenge Level:

Jane accidentally multiplied by 54 instead of 45, and her answer was 198 too big. What number did she multiply 54 by?

##### Age 11 to 14 Short Challenge Level:

Each interior angle in a quadrilateral (apart from the smallest) is twice the previous one. What is the size of the smallest interior angle?

### To Run or Not to Run?

##### Age 11 to 14 Short Challenge Level:

If an athlete takes 10 minutes longer to walk, run and cycle three miles than he does to cycle all three miles, how long does it take him?

### Magic 7

##### Age 11 to 14 Short Challenge Level:

Can you complete this magic square with the numbers from 7 to 15?

### Named Products

##### Age 11 to 14 Short Challenge Level:

Can you find the solution to this equation? Each of the different letters stands for a different number.

### Table Total

##### Age 14 to 16 Short Challenge Level:

Can you find the sum of all of the numbers in the table?

##### Age 14 to 16 Short Challenge Level:

In a 7-digit numerical code, each group of four consecutive digits adds to 16, and each group of five consecutive digits adds to 19. What is the sum of all 7 digits?

### Symmetric Reasoning

##### Age 14 to 16 Short Challenge Level:

Can you sum up the solutions to this equation?

### In the Hundreds

##### Age 14 to 16 Short Challenge Level:

For how many positive values of $n$ are both $\frac n2$ and $2n$ three-digit whole numbers?

### Pure Algebra

##### Age 14 to 16 Short Challenge Level:

Can you work out the product of $a$ and $b$ from the information in the question?

### Bat and Ball

##### Age 14 to 16 Short Challenge Level:

Can you work out how much this bat costs?

### Weighing the Baby

##### Age 14 to 16 Short Challenge Level:

Weighing the baby at the clinic was a problem. Can you work out the total weight of the baby, the nurse and me from the information given?

### I Love Musical Stars

##### Age 14 to 16 Short Challenge Level:

Each row and column has the total given. What is the value of the heart symbol?

### Day of the Triffids

##### Age 14 to 16 Short Challenge Level:

Jasmine buys three different types of plant. How many triffids did she buy?

### Weighing Scales

##### Age 14 to 16 Short Challenge Level:

Can you work out how many spheres will balance a single pyramid?

### Square to a Rectangle

##### Age 14 to 16 Short Challenge Level:

Kevin has moved some tiles to change the shape of his patio from a square to a rectangle. What are the lengths of the sides of the rectangle?

### Fractions of Fractions

##### Age 14 to 16 Short Challenge Level:

If you know that a fraction of X is the same as a different fraction of Y, can you work out X/Y?

### Near 10

##### Age 14 to 16 Short Challenge Level:

10 must remain within easy reach...

### Average Surroundings

##### Age 14 to 16 Short Challenge Level:

Can you work out Ali's age based on the diagram?

### Black and White Socks

##### Age 14 to 16 Short Challenge Level:

There are 20 black socks and some white socks in a drawer. Can you work out how many of the socks are white?

### Table Height

##### Age 14 to 16 Short Challenge Level:

A table has blocks of wood placed on and next to it. Can you work out how tall it is?

### Packing Boxes

##### Age 14 to 16 Short Challenge Level:

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

### Thousands of X

##### Age 14 to 16 Short Challenge Level:

Can you solve this scary-looking equation?

### Big Fish

##### Age 14 to 16 Short Challenge Level:

Granny has taken up deep-sea fishing! Last week, she caught a fish so big that she had to cut it into three pieces in order to weigh it...

### Oldest and Youngest

##### Age 14 to 16 Short Challenge Level:

Edith had 9 children at 15 month intervals. If the oldest is now six times as old as the youngest, how old is her youngest child?

### Sum = Product = Quotient

##### Age 14 to 16 Short Challenge Level:

Can you find a pair of numbers such that their sum, product and quotient are all equal? Are there any other pairs?

### Traffic Tunnel

##### Age 14 to 16 Short Challenge Level:

Will these vehicles fit through this tunnel?

### Simultaneous Multiplying

##### Age 14 to 16 Short Challenge Level:

If $a \times b=2$, $b \times c=24$, $c \times a=3$ and $a$, $b$ and $c$ are positive, what is the value of $a+b+c$?

### Relative Angles

##### Age 14 to 16 Short Challenge Level:

Can you work out the angles of a triangle from the relationships between them?

### Alien Currency

##### Age 14 to 16 Short Challenge Level:

The planet Zog has both green and blue bank notes. Can you work out how many zogs two green and three blue bank notes are worth?

### Tennis Training

##### Age 14 to 16 Short Challenge Level:

After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?

### Reciprocal Values

##### Age 14 to 16 Short Challenge Level:

Manipulate one algebraic fraction to solve another.

### Monetary Difference

##### Age 14 to 16 Short Challenge Level:

Al, Bertie, Chris and Di have sums of money totalling £150... What is the difference between the amount Al and Di have?

### Algebraic Differences

##### Age 14 to 16 Short Challenge Level:

If $6x−y=21$ and $6y−x=14$, what is the value of $x−y$?

##### Age 14 to 16 Short Challenge Level:

If you know how many books each boy, girl and teddy bear borrowed from the library, can you work out the number of girls?

### Three Fruits

##### Age 14 to 16 Short Challenge Level:

Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?

### A Third of the Area

##### Age 14 to 16 Short Challenge Level:

The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?

### Nested Square Roots

##### Age 14 to 16 Short Challenge Level:

Can you find the value of this expression, which contains infinitely nested square roots?