Question Video: Finding Four Proportional Numbers given the Relations between Them

Determine the four proportional numbers of which the fourth proportional equals the square of the second, the first is less than the second by 1, and the third equals 72.

03:20

Video Transcript

Determine the four proportional numbers of which the fourth proportional equals the square of the second, the first is less than the second by one, and the third equals 72.

Let’s set up what we know. We have one and two and then three and four. We know that the fourth proportional equals the square of the second. Let’s call the second 𝑏. That would make the fourth 𝑏 squared, since the fourth is the square of the second. The next thing we’re given is that the first is less than the second by one. The first value would be equal to 𝑏 minus one, and the third value equals 72. We can take these proportions and set them up as equivalent fractions. 𝑏 minus one over 𝑏, how do we get from 𝑏 to 𝑏 squared. Well we multiply by 𝑏, and that means that 𝑏 minus one times 𝑏 must be equal to 72. If we set this up, 𝑏 times 𝑏 minus one equals 72, we’ll be able to solve for 𝑏.

First we distribute the 𝑏. 𝑏 times 𝑏 is 𝑏 squared minus 𝑏 equals 72. If we move the 72 to the other side of the equation, we’ll get 𝑏 squared minus 𝑏 minus 72 equals zero. And from there we can factor to find out what 𝑏 would be. We need two numbers that multiply together to equal 72 and add together to equal negative one. I know that eight times nine equals 72. And if we have a negative nine and a positive eight, they add together to equal negative one. 𝑏 plus eight is set equal to zero, and 𝑏 minus nine is set equal to zero. 𝑏 equals negative eight or 𝑏 equals positive nine. And so we need to consider both cases.

On the left we’ll show 𝑏 equals negative eight and on the right 𝑏 equals nine. Here are our four options: the first one is 𝑏 minus one. Negative eight minus one is negative nine. 𝑏 equals negative eight. Our third value equals 72, and our fourth value equals 𝑏 squared. Negative eight squared equals 64. The values negative nine, negative eight, 72, 64 fit these requirements. And now we need to plug in nine for 𝑏. 𝑏 minus one equals eight. 𝑏 equals nine. Then we have 72. And finally 𝑏 squared equals 81. The values eight, nine, 72, and 81 also fit the above statements.

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