Sophie’s baby weighed 3200 grams correct to two significant figures. Part a) one) Write the smallest possible weight of the baby.
The first significant figure in a number is the first nonzero digit. The second significant figure is the second digit after the first significant figure — that, in this case, is the two. The weight of Sophie’s baby has been rounded to two significant figures. The third digit in this number would have been the deciding digit. When the deciding digit is five or above, we round the number up. And when the deciding digit is less than five, we round down.
To find the smallest possible weight of the baby, we need to find the lowest possible weight that would still round up to 3200 grams. Since the deciding digit that’s the third digit needs to be five or above, the smallest possible weight of the baby that would still round up to 3200 is 3150.
Part two) Write the largest possible weight of the baby.
Once again, let’s consider the third digit, the deciding digit. We want to find the largest possible weight that will still round down to 3200. For the number to still round down to 3200, this digit must be less than five. In fact, it’s four. However, 3240 is not the largest possible weight that would still round down to 3200.
3249 grams would still round down as would 3249.9 as indeed would 3249.99. And we can continue this pattern. The number would get closer and closer to 3250; it will never quite get there. The largest possible weight of the baby then is 3249.99999 and so on grams. We could write that as 3249.9 recurring.
Now, it’s important to differentiate between this and the upper bound of the weight. We say that the upper bound is 3250 grams. The weight could not actually be 3250 grams, but it gets so close. So we call that the upper limit. Since this is just asking us the largest possible weight, we call that 3249.9 recurring grams.
Over the course of a month, Emily’s baby’s weight increase by seven percent. At the end of the month, the baby weighed 3210 grams. Calculate the weight of the baby at the beginning of the month.
We cannot just subtract seven percent from the new weight of the baby. But there are two different ways that we can calculate the weight of the baby at the beginning of the month — that’s before the increase of seven percent.
Let’s consider this first method. This method can be used in both the calculator paper and a non-calculator paper. Emily’s baby’s weight has increased by seven percent. We can say that the baby originally weighed 100 percent of its weight. When we increase that by seven percent, we added on and that gives us 107 percent. That means that 107 percent of the weight of the baby is equal to 3210 grams.
To find the baby’s original weight or the weight at the beginning of the month, we need to calculate what 100 percent is. To do this, we first calculate the value of one percent. To get from 107 percent to one percent, we divide by 107. We must do the same to 3210. 3210 divided by 107 is 30. So one percent of the baby’s original weight is 30 grams. Once we know what one percent of the weight is, we can multiply this by 100. One multiplied by 100 is 100 percent and 30 multiplied by 100 is 3000.
So the baby weighed 3000 grams at the beginning of the month.
The alternative method is to form an equation. Let’s call the weight of the baby at the beginning of the month 𝑥 or 𝑥 grams. We said since the baby’s weight had increased by seven percent, it was now worth 107 percent of the original weight. Percent means out of 100. So 107 percent is the same as 107 over 100. This is also equivalent to 1.07 since when we divide by 100, we move the digits to the right two places.
So if we knew the original weight of the baby, we can multiply it by 1.07 to get the new weight. We don’t though; we’ve called the original weight of the baby 𝑥. So when we multiply 𝑥 by 1.07, we get the new weight, which we said was 3210 grams.
To solve for 𝑥 and calculate the original weight of the baby, we can divide both sides of this equation by 1.07 and that will tell us that 𝑥 is equal to 3210 divided by 1.07, which is once again 3000.
The baby weighed 3000 grams at the beginning of the month.