The Fraction Inside Your Heart

More than 400 years ago, physician William Harvey faced a problem: the accepted explanation of blood flow simply did not add up. By estimating how much blood the heart moved with every beat—and multiplying it by the number of beats in a minute—he showed that the body could not possibly be making and consuming that much new blood. The blood had to be circulating.

In this first episode of Story Problem, we use Harvey’s discovery to explore one of the most important ideas in mathematics: fractions and percentages.

You’ll learn what doctors mean by ejection fraction, the percentage of blood the heart pumps out with each contraction. Two hearts can pump out exactly the same amount of blood and still have very different ejection fractions because they began with different amounts.

Along the way, we break every percentage problem into three pieces:

Part = Percent × Whole

You’ll practice finding:

• The part when you know the percent and the whole
• The percent when you know the part and the whole
• The whole when you know the part and the percent

You’ll also see how stroke volume and heart rate combine to produce cardiac output—and how a simple multiplication helped overturn more than a thousand years of medical thinking.

Just listen, follow the story, and try the problems in your head.

Story Problem — Science and Math Made Clear...Through Stories.

Transcript

SPEAKER_00 0:11

Imagine you're in Egypt 3,000 years ago. You're an embalmer, and it's your job to prepare people for their journey into the next world. And along the way, you've learned more about the human body than almost anyone else. You've seen and held the organs, including the heart. You know there are vessels running out of it, threading throughout the whole body. But here's what you can't answer: where does the blood come from? And once it flows out into the body, where does it go? The Greek and Romans went several steps further. Hippocrates and Galen understood that arteries and veins were different things, but still nobody could answer the most basic question. Where does the blood come from and where does it go? Galen had an answer. He believed the liver was constantly making new blood out of food. That blood would travel outward through the veins, into the arms and legs, and the rest of the body. And once it got there, the tissues would use it up, almost like fuel being burned. And honestly, it's not a ridiculous idea. It explains what you can see. You eat food, the body makes something from it, blood moves outward, the tissues need nourishment. It made enough sense that people believed it for nearly 1400 years. Then in 1628, an English physician named William Harvey published a very small book that completely changed our understanding of the heart. And he didn't do it with a better microscope. Harvey never saw capillary. Those tiny vessels connecting the arteries and the veins wouldn't be seen until four years after his death. Instead, Harvey took Galen's idea apart with arithmetic. Harvey came from a family of merchants. He grew up around the language of trade, money going out, money coming in, and ledgers that had to balance. Even the word he eventually used, circulation, was already connected to money moving around and coming back again. And when Harvey faced a question that nobody could answer by looking, he didn't guess. He counted. He didn't know exactly how much blood the heart pushed out with each beat, so he deliberately chose a very small estimate. The smallest amount he thought the heart could possibly be pushing out. Then he multiplied. A small amount of blood per beat times the number of beats in a minute times 30 minutes. And the number he got was enormous. In only half an hour, the heart would move more blood than the entire body contained. Over an hour, more than the person's entire body weight. Now think about what that would mean if Galen were right. The liver would have to manufacture hundreds of pounds of brand new blood every day from the food the person wasn't even eating. That couldn't be happening. The numbers didn't balance. So the blood could not simply travel outward and disappear. It had to be coming back. The heart wasn't constantly making new blood. It was moving the same blood around and around the body. Blood circulated. And Harvey reached that conclusion not because he saw the whole system, because he did the math. The pattern was simple. Amount of blood per beat times beats per minute equals the total the heart pumps out in a minute. Not a new machine, not a complicated equation, just careful observation and multiplication. Hold on to that pattern. We'll use it again before we're done. This is Story Problem. I'm Beheer Bani. I'm making this show for parents and adult learners who maybe don't feel like math people, who might carry some old anxiety from math or science classes or from standardized tests, but who still want to understand something real without feeling like they can't catch up. We don't have any beakers on this show, no slime, no forced enthusiasm, just stories that are actually true and math that's actually real. Episode one is about a number called the ejection fraction. And by the end, you're going to know all three ways a percent problem can be asked and be able to solve each one out loud using how your own heart works as an example. Harvey figured out that blood moves in a loop, but there was still another question he couldn't answer. Every time the heart beats, how much blood inside of it actually gets pushed out? Today we have a number for that. And as a cardiologist, I use this number every single day. It's called the ejection fraction. Your heart does not squeeze itself completely empty with every beat, and it and it isn't supposed to. A healthy heart usually puts out somewhere between 55 and 70% of the blood inside its main pumping chamber, the left ventricle. The rest stays behind, waiting for the next beat. Imagine the heart fills with 100 milliliters of blood. If it pushes out 60 milliliters, its ejection fraction is 60%. That's really all the term means. Ejection is the amount pushed out. Fraction tells us we're comparing that amount with how much was there to begin with. So the question is: out of all the blood that could have left, what fraction actually did? When that number falls well below normal into the range we call heart failure with reduced ejection fraction, it tells us the heart's squeeze has weakened. It's an extraordinarily useful measure of the heart's pumping strength. So here's the concept. Ejection fraction is the fraction of blood inside the heart that gets pumped out with each beat. Amount pushed out divided by the amount that was there in the beginning. That's it. But that math can be asked of us in three different ways. Sometimes we know the total and the percentage, and we find how much came out. Sometimes we know how much came out and how much was there, and we find the percentage. And sometimes we know how much came out and what percentage that was, and we work backward to find how much was there in the first place. Those are the three kinds of percentage problems. Once you can tell which one you're being asked, the math gets much easier. Let's do the first kind together. Stay with me, even if you're driving or you don't have anything to write down with, you don't need to jot anything down, just follow the numbers in your head. The heart fills with 120 milliliters between beats. That 120 is the hole, the total sitting in the chamber. The ejection fraction is 60%, which just simply means 60 out of every 100. If the heart starts with 120 and pumps out 60%, how many milliliters actually leave? We know the hole, we know the percent, we're looking for the part. What is 60% of 120? Here's the easy way to figure that out in your head. 10% of 120 is 12. Multiplied by 6, 12 times 6 is 72. So the heart pumps out 72 milliliters. You knew the whole, you knew the percent, and you solved for the part. This pattern shows up everywhere. And on a test, it's often hiding inside a store discount. Say a jacket costs $80 and it's 25% off. 25% is one quarter, and one quarter of 80 is $20. That's the discount. But watch the question. It usually wants the sale price, not the discount. So 80 minus 20 is 60. The jacket costs $60. The word problem changes, but the math stays the same. Today the percentage described blood leaving the heart. Tomorrow it might be a sale price, a test score, or the growth of a population. But finding the part is only one of three ways a percentage problem can be built. Let's meet the other two. The second kind, we know the part and the whole, and we want the percent. Imagine two hearts. Heart A fills with 110 milliliters and pumps out 70. Heart B fills with 200 and also pumps out 70. Same amount out, 70 milliliters each. At first glance, they look equal, but they're not. Heart B first because it's easier. Let's figure that one out. 70 milliliters out of 200. We to figure this out, we have both numbers, so 35 out of 100, and that's 35%. Now heart A, 70 out of 110. The numbers don't work out quite as well, but let's estimate first. 70 out of 100 would be 70%, but it started with a little more, 110 milliliters, so it's a little less than that. Divide it out, and it's about 64%. So heart A is 64%, heart B is 35%. Same 70 milliliters out, completely different meaning. As a cardiologist, 64% is normal and healthy, that's a normal and healthy ejection fraction. 35% gets my attention. That heart's pumping function is reduced. And here's the reason: the top number, the numerator, was identical, 70 in both. But the totals they started from were completely different. That's the lesson to carry out of today as well. The same number can mean very different things depending on what it's measured against. The third kind, we know the part and the percent, and we're missing the whole. A heart pumps out 70 milliliters, and that's 30% of what was inside before it squeezed. How much was there to begin with? This one works backward. Before we knew the whole and multiplied, now we know the part and divide. 70 divided by 0.35 is 200 milliliters. Let's take a quick check of that. 35% is a little bit more than a third, so the whole should be a little under 3 times 70. 3 times 70 is 210, and we got 200, right where we'd expect. And notice when the percent is under 100, the whole is always bigger than the part. If your answer ever comes out smaller, you know something went wrong. All three are really the same relationship rearranged. The part equals the percent times the whole. Missing the part, multiply. Missing the percent, divide the part by the whole. Missing the whole, divide the part by the percent. Same three ingredients every time. A part, a whole, and a percent, connecting them. So when you see a percentage problem on a on a test, in a store, or on an echocardiogram, the first question is never what formula do I use. It's which of the three pieces am I missing? Once you know that, most of the problem is already solved. One last trick for word problems is means equals of means times. 70 is what percent of 200 becomes 70 equals some percent times 200. The equation is already sitting there. One more calculation, and it brings us back to Harvey. Our heart pumps 72 milliliters per beat. Say it beats 70 times a minute. How much in one minute does it does the heart put out? This is the same pattern Harvey used. He used the amount per beat times beats equals the total amount out. So in our case, 72 times 70. Uh, to make this simpler, 72 times 7 is 504, and then we add a zero to that for the zero we took away from 70. That's 5040 milliliters, just over five liters. So think about that. Sitting still doing really nothing, your heart pumps roughly your entire blood supply every minute, which is about five liters. Not five liters of new blood, the same blood coming back again. That's what we call the cardiac output. And we're right back where we began. Harvey never saw the capillaries, but the numbers told him that the blood had to be coming back. Now let's try each one of the different problems. I'll give the problem then pause, and you try it first. Just ask, what do I know and what am I looking for? Here's the first one. A heart fills with 100 milliliters and has an ejection fraction of 55%. How much pumps out? When the hole is 100, this one's almost automatic. 50% 55% of 100 is 55 milliliters. Here's the second problem. A heart fills with 150 milliliters and pumps out 45. What's the ejection fraction? We know the part and the whole, we want the percent. 10% of 150 is 15, and 45 is three times 15, so 30%. That's a reduced ejection fraction. Problem number three, a heart pumps out 60 milliliters, and that's 40%. What's the whole? We know the part and the percent, and we want the whole. If 40% is 60, then 10% is 15, so 100% is is 150 milliliters. Let's check that. 40% of 150 is 60, so that works. So here's a whole episode summarized in one idea. Your heart doesn't empty completely when it beats, it pumps out only a fraction of the blood inside, and that fraction is the ejection fraction. Percentage problems work the same way. There's always a part, a whole, and a percent, and the only real question is which piece you're missing. Well, this has been Story Problem. Thanks for spending this time with me. And if you're worked through those problems along the way, no pencil, no paper, no calculator, you just proved you could do real math anywhere. If you enjoyed this episode, please come back for future episodes. Subscribe, and the best thing you could do is recommend this show to a friend. Thank you so much.