How does 5.9-7.66=-1.76?

Question

Sorry for such an easy question but this has been confusing me like crazy. I have borrowed like normal and subtracted like every other problem, and the answer I get is -2.24. How is this wrong?

Answer 1

5.9+0.1=6

6+1=7

7+0.66=7.66

0.1+1+0.66=1.76

why can we do it this way?

because 5.9-7.66=x can be written as

5.9-x=7.66

since 7.66>5.9, -x has to be positive, so x has to be negative, therefore the solution is -1.76.

Answer 2

$\require{cancel}$ \begin{array}{ll} &5.\quad\cancelto{8}{9}\,\cancelto{10}{0}\\ -\!\!\!&7.\quad6\quad6\\\hline -\!&2.\quad2\quad4 \end{array}

which is incorrect. So I don't think "borrowing" works when we subtract a "larger" (larger in magnitude) number from a smaller one (smaller in magnitude).

Instead, consider their magnitudes, that is, the distance each number is away from the $0$. Notice that the magnitude of $-7.66$ is larger than the magnitude of $5.90$, so we know that the final answer will be negative. So, we subtract the smaller magnitude number from the larger magnitude number and remember to force the final answer to be negative. Then we proceed as usual \begin{array}{r lll} &\cancelto{6}{7}.&\cancelto{16}{6}&6\\ -& 5\quad.&9&0\\\hline -&1\quad.&7&6 \end{array} which is correct, $-1.76$

Answer 3

We have $5.9$. If you subtract one, you have $4.9$. Subtract $2$, you have $3.9$. Subtract $3$, $2.9$. Naively, one might expect that subtracting $7$ therefore gives you $-2.9$. This fails, however, because this "pattern" breaks down once you start going into the negatives. $0.9 - 1 = -0.1$. The pattern reemerges, so now $-1.1$; we have subtracted $7$ from $5.9$.

At this point, we deal with the decimal component. It's easy to see that the final answer is $-1.76$.

Answer 4

If you consider the parts before and after the decimal point separately, you have this: \begin{align} 5.9 &= 5 + .9 \\ 7.66 &= 7 + .66 \\ \end{align} If you subtract the $7$ from the $5$ on one hand, and $0.66$ from $0.9$ on the other hand, you get \begin{align} 5 && .9\phantom{0} && \tag1\\ \underline{-7} && \underline{-.66} \tag2\\ -2 && .24 \tag3\\ \end{align} The key thing here is, what does the last line mean? On line $(1)$, the sum of the numbers in the two columns, $5$ and $0.9$, is one of the numbers we started with, $5.9$, while on line $(2)$, the sum of the numbers in the two columns, $-7$ and $-0.66$, is $-7.66$, corresponding to the operation of subtracting $7.66$. On line $(3)$, then, we must not subtract $0.24$ from $-2$ in order to obtain $-2.24$; rather, we must add the two numbers, like this: $$ (-2) + 0.24 = -(2 - 0.24) = -1.76. $$

In practice, this is still too confusing, and a better method is to rearrange the original subtraction so that when you do the digit-by-digit calculation you are always doing it with the larger-magnitude number on top. So we first write $$ 5.9 - 7.66 = - (-5.9 + 7.66) = -(7.66 - 5.9). $$ \begin{align} 7 &.66 \\ \underline{-5} & \underline{.9\phantom{0}} \\ 1 &.76 \end{align} Therefore $$ 5.9 - 7.66 = -(7.66 - 5.9) = -1.76. $$