No Calculation Required! Trivia Quiz

Answer: Too small

The price in the sale must be less than the original price, hence the answer (the original price) must be larger than $40.

As for the calculation, 80% of the price is $40 so a number of different methods, most concisely 40/0.8, lead to an original price of $50.

Answer: 8 squares

A 3x2 rectangle of 6 squares is completely within the trapezium, so its area is more than 6. On the other hand, the trapezium is contained within a 5x2 rectangle of 10 squares, so it is less than 10 squares. Its area must be the only option strictly between these areas, 8 squares.

As for the calculation, we could start from 6 or 10 squares and add or subtract the triangle areas that make up the parts of squares included in the area. We could also use a specific formula for the area of a trapezium, like 0.5*(3+5)*2 = 8 squares.

Answer: x = -7

If you add a positive number to 44, you won't get 9. This means five times the unknown number is negative. Therefore, the answer has to be the only negative option.

Indeed, multiplying -7 by 5 gives -35, and adding 44 gives the desired value of 9. This is called substitution and can also be used to rule out all the positive answer options.

Solving an equation involves reversing the process of substitution: 5x = -35 so that when we add 44 we get 9. Therefore, x = -7 so that when we multiply by 5 we get -35.

Answer: 4

Dividing by a number larger than 1 will make a value smaller (for instance, dividing by 2 halves a value), so the value is less than 4.8. As all values are positive, the answer will be positive. 4 is the only positive option less than 4.8.

As for the calculation, the short division correctly calculates 4.8 divided by 12, not 1.2. By comparing the calculations, the answer should be 10 times larger than 0.4. You could instead see that 4.8/1.2 = 48/12 are equivalent fractions or that 1.2+1.2+1.2+1.2 = 4.8 so 1.2*4 = 4.8.

Answer: Too large

A probability must be between 0 and 1. If measured in percentages, this would be between 0% and 100%. When there are finitely many outcomes, like here (0, 1, 2, 3, 4 or 5 heads), 0 is the probability of an impossible event and 1 is the probability of a certain event. (Stranger things happen if there are infinitely many outcomes.)

As for the calculation, the number of ways to choose 3 out of 5 items needs to be divided by 3! = 6, the number of symmetries as it doesn't matter in what order the 3 heads occur. This means the true probability is 0.3125.

Answer: 30 minutes

It should take less time with more teachers working on the task. The only choice less than 60 minutes is 30 minutes. Specifically, the number of teachers has doubled, so the time taken should halve.

We have to make a number of assumptions to say that the task will take 30 minutes. The teachers might have some economy of scale, so they are able to mark faster as a collective, or they might get in each other's way. They might work more or less effectively as time goes on. All of these factors are ignored through the phrase "working at a constant rate".

Formally, the two variables - teachers and time taken - are inversely proportional. It takes 120 minutes of labor to complete the task, which could be divided by 4 teachers equally into 30 minutes each.

Answer: 60

60 degrees is the only acute angle listed - the angle smaller than a right angle. Grab a piece of paper and try to draw a triangle with two right angles. You need more than three lines, or a curved line, or some other violation of what a triangle is. In 2D space, the angles in a triangle add up to 180 degrees, so at least two of its three angles are acute.

As for the calculation, 2 m is the hypotenuse (the longest side, opposite the largest angle) and 1 m is adjacent to the angle x. This means that cos(x) = 1/2 and so x is 60 degrees.

Answer: 10

All the answers other than 10 are too big: 100 is 50*2, which should be much bigger than 50*0.2, and the others are even larger. Multiplying 50 by a number less than 1 should give us a number less than 50.

The image shows that it takes 5 lots of 0.2 to make a 'whole' i.e. 5*0.2 = 1. Then, with 50 lots of 0.2, we can make 10 'wholes' i.e. 50*0.2 = 10.

Another way to compare is by saying that 50*0.2 should be 10 times smaller than 50*2.

Answer: 163 cm

The mean is an average, which means it should be somewhere between the smallest and largest values. As the top and bottom rows are empty, everyone surveyed was between 150 cm and 180 cm and so the only number in that range is 163 cm.

As for the calculation, the mean can only be estimated since the full data is not available (the two people between 170 cm and 180 cm could - for instance, be 170 cm tall, 179 cm tall or 175 cm tall). A sensible estimate assumes all 10 people are either 155 cm, 165 cm or 175 cm tall (depending on their row), leading to the calculation (5*155+3*165+2*175)/10 = 162 cm (close to the true value!).

Answer: No

All three rectangles are congruent - the same size and shape - so they have the same area i.e. space inside of them. Changing which labels are on the shape doesn't change the area: in this case 16 square centimeters.

However, if you "multiply all the sides together" without care, you will get 3 different answers.

Even if you take "all sides" to mean "all four sides of the rectangle" then Tommy's rule is not correct. The area of a rectangle is its base multiplied by its height - in this example 3*4 = 12 square cm. Specifically, it is the 2 different side lengths that need to be multiplied together.