Equations and Formulae - Short Problems
Luis' Eight
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
Given some relationships amongst these shapes, how many triangles equal one diamond?
Seven Dwarfs
The Seven Dwarfs have seven different birthdays. How old can they be?
Five-to
The number 2005 is the sum of a sequence of five consecutive positive integers. What is the smallest of these integers?
Partial Magic
This magic square has only been partially completed. Can you still solve it...
Interstellar
What number could replace * so that */5 is between 3 and 4?
Turnips
Baldrick could buy 6 parsnips and 7 turnips, or 8 parsnips and 4 turnips. How many parsnips could he buy?
Crossover
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
You may have seen magic squares before, but can you work out the missing numbers on this magic star?
Multiplication Mistake
Jane accidentally multiplied by 54 instead of 45, and her answer was 198 too big. What number did she multiply 54 by?
Geometric Quadrilateral
Each interior angle in a quadrilateral (apart from the smallest) is twice the previous one. What is the size of the smallest interior angle?
Fifty Coins
Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?
Symbol
Using the new operator $\oplus$, can you solve this equation?
To Run or Not to Run?
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
Can you complete this magic square with the numbers from 7 to 15?
Named Products
Can you find the solution to this equation? Each of the different letters stands for a different number.
Adjacent Additions
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
Can you sum up the solutions to this equation?
In the Hundreds
For how many positive values of $n$ are both $\frac n2$ and $2n$ three-digit whole numbers?
Pure Algebra
Can you work out the product of $a$ and $b$ from the information in the question?
Weighing the Baby
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
Each row and column has the total given. What is the value of the heart symbol?
Table Total
Can you find the sum of all of the numbers in the table?
Weighing Scales
Can you work out how many spheres will balance a single pyramid?
Square to a Rectangle
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
If you know that a fraction of X is the same as a different fraction of Y, can you work out X/Y?
Average Surroundings
Can you work out Ali's age based on the diagram?
Black and White Socks
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
A table has blocks of wood placed on and next to it. Can you work out how tall it is?
Packing Boxes
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.
Big Fish
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
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
Can you find a pair of numbers such that their sum, product and quotient are all equal? Are there any other pairs?
Simultaneous Multiplying
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
Can you work out the angles of a triangle from the relationships between them?
Alien Currency
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
After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?
Reciprocal Values
Manipulate one algebraic fraction to solve another.
Monetary Difference
Al, Bertie, Chris and Di have sums of money totalling £150... What is the difference between the amount Al and Di have?
Algebraic Differences
If $6x−y=21$ and $6y−x=14$, what is the value of $x−y$?
Well Read
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
Quince, quonce and quance are three types of fruit. Can you work out the order of heaviness of the fruits?
Day of the Triffids
Jasmine buys three different types of plant. How many triffids did she buy?
A Third of the Area
The area of the small square is $\frac13$ of the area of the large square. What is $\frac xy$?
Nested Square Roots
Can you find the value of this expression, which contains infinitely nested square roots?