Consider the multiplication grid shown below. The product of the numbers in a single row, column, or diagonal is one. Complete the grid.
The word “product” means multiply. For example, the product of three and five is 15, as three multiplied by five is equal to 15. We’re told in this question that the product of each row, column, and diagonal equals one. Before starting, it is worth considering which pairs of numbers have a product of one.
Any number and its reciprocal have a product of one. For example, four and its reciprocal, one-quarter, have a product of one. Four multiplied by one-quarter equals one. We can see this as four is the same as four over one. Multiplying the numerators gives us four, and multiplying the denominators also gives us four. Four divided by four equals one.
This also works with fractions. The reciprocal of two-fifths is five over two or five-halves. We flipped the fraction. The numerator becomes the denominator, and the denominator becomes the numerator. Multiplying the numerators gives us 10, and multiplying the denominators also gives us 10. 10 divided by 10 is equal to one. Therefore, two-fifths multiplied by five-halves equals one.
Let’s see how we can use this fact to help us work out the missing values labelled 𝑎, 𝑏, and 𝑐. Let’s first consider the top row: 𝑎, 40, and one-tenth. As the product of these three numbers equals one, 𝑎 multiplied by 40 multiplied by one-tenth is equal to one. 40 multiplied by one-tenth is equal to four, as 40 divided by 10 is four. This gives us 𝑎 multiplied by four is equal to one.
As already mentioned, any number and its reciprocal have a product of one. The reciprocal of four is one-quarter. This means that 𝑎 is equal to one-quarter. Another way of doing this final step would be to divide both sides of the equation 𝑎 multiplied by four equals one by four. One divided by four is a quarter, and 𝑎 multiplied by four divided by four is equal to 𝑎. The top left number in our grid is one-quarter, as one-quarter multiplied by 40 multiplied by one-tenth equals one.
Let’s now consider the second row. Two-fifths multiplied by 𝑏 multiplied by five-halves is equal to one. Let’s multiply the two fractions first. We’ll multiply two-fifths by five-halves. Two-fifths and five-halves are the reciprocal of each other. And we’ve already seen that they multiply to give us one. Our equation simplifies to 𝑏 multiplied by one is equal to one.
Any number or letter multiplied by one is equal to itself. Therefore, 𝑏 equals one. The middle number in our grid is one, as two-fifths multiplied by one multiplied by five-halves equals one. To work out the final missing number 𝑐, we could either use the bottom row, the right-hand column, or the diagonal. This time, we will use the bottom row. However, we would get the same answer if we use the diagonal or the right-hand column.
The bottom row gives us 10 multiplied by one fortieth multiplied by 𝑐 is equal to one. Our first step is to multiply 10 by one fortieth. 10 is the same as 10 over one. Multiplying the numerators gives us 10, and multiplying the denominators gives us 40. This fraction can be simplified to one-quarter by dividing the numerator and denominator by 10. Our equation is therefore one-quarter multiplied by 𝑐 is equal to one.
We could multiply both sides of this equation by four. This would cancel the quarter and give us our value of 𝑐. Alternatively, the reciprocal of one-quarter is four. Therefore, one-quarter multiplied by four must be equal to one. The final missing answer in our grid, labelled 𝑐, is equal to four.
We now have the three missing values in the grid. They are one-quarter, one, and four. We can check these answers by multiplying any of the rows, columns, or diagonals and ensuring that the product is equal to one.