Let’s try other word problems. This first is a classic. The Brit has $ 2 and twenty-five cents in fives and dimes. If he has a total of 40 coins, how many has from each value? First, let’s define the variables. The letter n will indicate the number of dimes. … And the letter d the number of fives. We know he has a total of $ 2 and 25 cents. in dimes and fives. How many coins is it from each value? So if we take the number of fives, everyone has value of $ 0.05. So the number of fives times 0.05 is how much money represent fives, plus the amount of money representing dimes. So we have d dimes. Each is worth $ 0.1. So that’s the total amount of money we have. If we have 5 fives, we multiply 0.05 by five and get 0.25. If we had 4 dimes, it would make 0.40 and we would add it up put it together. Our assignment says that the total amount we have, They are $ 2 and 25 cents. This first statement can be translated into this equation. We have a certain number of fives and a certain number of dimes. Multiply 0.05 by the number of dimes and 0.1 by the number of dimes and we get 2.25. Now the second statement – we have a total of 40 coins. And we assume they’re just dimes and dimes, no more coins. So a total of 40 coins, that is, the number fives plus the number of dimes must put together 40. So dimes plus dimes – equals 40. This is what this information tells us. So we have two equations with two unknowns and two ways to calculate it. We can construct a system of two equations and subtract one equation from the second. Or we can make a substitution. Let’s try it. This is the more common way. So – what do we do: solve the equation for d. We could work it out for any stranger, though we find out how much is d and what comes out we replace d here in this equation. First we subtract n from both sides of the equation – – Here we go. Minus n plus this, minus n. Of course, this has to be deducted from both sides and we’re left with that d equals 40 – n. Now let’s take the result and substitute it into the second equation. d equals 40 minus number of fives. And we can put it back in d. The first equation will now be 0.05 times number of fives plus 0.1 times number of dimes. So we already know the number of dimes equal to 40 minus n. We learned this from the second statement. So instead of d, I substitute 40 minus n. That’s the number of dimes we have. It’s 40 minus number of fives and that equals two dollars and twenty-five cents. And now we can try to solve it for n. So we get 0.05 times n plus 0.1 times 40 – that’s 4. See how we multiply it. 0.1 times 40, that is 4 minus 0.1 times n, That’s how I multiplied the parenthesis, and the whole thing equals two dollars twenty-five cents. And what do we do next? Now either take 0.05 and subtract 0.1 from it depending on how we want it – subtracting these two terms, we get to minus – 0.05 minus 0.1 – by subtraction we get a negative number plus 4 – this four still stays here and it is equal to 2.25 Now we can subtract 4 from both sides of the equation We get minus 0.05n equals (if we subtract 4 left – the four must be gone – subtract 4 even here; 2.25 minus 4 – that’s minus 1.75 OK. Now multiply both sides of the equation minus one, to simplify it, so we get out of it plus and plus. And then we divide both sides by 0.05. We could simply divide both sides by minus 0.05 but i like everything positive. So when we divide everything by 0.05, we get that n is equal to 1.75 broken 0.05. So let’s divide it. I’ll write it in a different color. So divide by 1.75 / 0.05. Let’s multiply first numerator and denominator stem. We move two decimal places, so we will have 5 and 175 It’s the same as 175 divided by 5. 17 divided by five are 3. 3 times five is fifteen 2 onwards We write 5. 25. 25 divided by 5 is 5. 5 times 5 is twenty five. We have no rest. So that’s equal to 35. n is therefore equal to 35 and we can find out how much d. d is equal to 40 minus n. d is thus equal to 40 minus 35, which is 5. So we have 5 dimes and 35 dimes. And to make sure it’s right, 35 fives gives how much? 35 times 5 cents is 175 cents or seventy-five cents a dollar. $ 1 and 75 cents in pennies. 35 times 5 or 0.05, depending on as you want. And so this must be 5. And this is going to be 50 cents in dimes. 1.75 plus 0.5 – that’s a total of two twenty-five cents. Let’s calculate one more word problem. We have square shapes here. We see them here. How many squares are in the twelfth figure? So let’s look at it. This is the first, this is the second, and this is the third pattern. We have a single square here, there are two times two, but this block is missing right here. So let’s look at it. One way to look at it is 2 times 2, and here 3 times 3, but from here 2 times 2 are taken out. So the next one will be 4 times 4 and the removed block will be 3 times 3. I would say that one way to describe the number of squares – – Actually, the easiest way, we don’t have to don’t even think. Let’s see, here we have three and then we have 3 minus 1 here. I’ll highlight it with a different color. 3 minus 1 here. Here we have 2 and here 2 minus 1. And here we have 1 and 1 minus 1. So zero. In general, we can say that the number of squares for any n will be n plus n minus 1. Which is 2n minus1. And it works. 2n – 1 2 times 3 minus 1 is 5 1, 2, 3, 4, 5. It works for these three shapes. So for the twelfth shape, where n equals 12, it will be 2 times 12 minus 1, ie 24 minus 1 which equals 23 squares in the twelfth figure. Although I said we would calculate only one more task, but let’s go to one more task, when we started like that. Grace rode her bike at 12 miles per hour. An hour later, Dan drove the same route at 15 miles per hour. How long will it take him before Will Grace arrive? So let’s denote the time Grace travels as t. Or rather t will be the time of Dan’s ride. How long will Grace’s ride take after that? Dan drove an hour later. So, whatever the number, Grace has to go an hour longer. So t plus 1 equals Grace’s travel time. When this is 0, this will be 1. She will already have a 1 hour drive. And we have to remember that distance is equal speed times time. So Dan’s distance will be 15 miles per hour times time in hours. That’s the distance Dan has come. And then Grace’s distance – we can call it g, or let’s say D for Grace, it’s also her distance traveled – – It’ll be 12 miles per hour. So her time will be t plus one. It’s one hour longer, than Dan. 12 times t plus 1. And the point at which they meet is the point in which both sides will equal. Dan’s distance traveled will be equal the distance Grace has traveled. So these two items must be equal. 15 times t – that’s how far Dan has come – – must be equal 12 times t plus 1. And that’s how far Grace went. And don’t forget, her time must always be an hour longer than Dan’s. So let’s count the time. We have 15t equals 12t plus 1. 12t plus 1. And then we subtract 12 – sorry 12t plus 12 – we don’t want to make a mistake here. When we subtract 12 on both sides of the equation, we get 3t equals 12. Divide both sides by three – move it a little lower – And we find that t equals 4. So after 4 hours, after driving 4 hours, caught up with Dan Grace. This is Dan, and if his journey took 4 hours, Grace had to drive 6 hours. 6 o’clock is the time Grace drove. And Dan after 4 hours – 4 times 15 ran 60 mil. And Grace – I’m sorry, she left 1 hour earlier, so that’s it not six o’clock, she only drove one hour longer than Dan. So it’s 5, she drove a total of 5 hours. Grace pedaled for 5 hours at 12 miles per hour. And she drove, once again, also 60 miles. So they met exactly after sixty miles. ..