# Significant figures of the product [closed]

How is

$$\frac{8.925 − 8.904}{8.925} × 100 = 0.24?$$

When I solved I got

$$\frac{0.021}{8.925} × 100$$

Now it will have to be $$2$$ significant numbers $$(0.021)$$ time $$1$$ significant number $$(100)$$ and the multiplication rule tells us that the answer will be of $$1$$ significant number, right?

But how come we ended up with $$0.24?$$

• I'm voting to close this question as off-topic because your question lacks context: This question needs revision before it is ready for a great answer. Please edit to include how this question came up and how you tried to answer it. This will help writing an answer that is useful for you and for others. – Karsten Theis Jun 13 '19 at 3:28
• In fact, the constant 100 in percentage calculation context has infinite number of significant digits, as 0 is significant digit as well, so 100=100.0000000000000.... – Poutnik Jun 13 '19 at 9:12

According to Columbia University Website, Rules for Significant Figures are as follows:

1. All non-zero numbers ARE significant: The number 33.2 has three significant figures because all of the digits present are non-zero.

2. Zeros between two non-zero digits ARE significant: The number 2051 has four significant figures. The zero is between two non-zero digits, 2 and 5.

3. Leading zeros are NOT significant: They're nothing more than "place holders." The number 0.54 has only two significant figures while 0.0032 also has two significant figures. All of the zeros are leading.

4. Trailing zeros to the right of the decimal ARE significant: For example, there are four significant figures in 92.00. Keep in mind that 92.00 is different from 92: A scientist who measures $$\pu{92.00 mL}$$ knows his value to the nearest 1/100th milliliter; meanwhile his colleague who measured $$\pu{92 mL}$$ only knows his value to the nearest $$\pu{1 mL}$$. It's important to understand that "zero" does not mean "nothing." Zero denotes actual information, just like any other number. You cannot tag on zeros that aren't certain to belong there.

5. Trailing zeros in a whole number with the decimal shown are significant: Placing a decimal at the end of a number is usually not done. By convention, however, this decimal indicates a significant zero. For example, "540." indicates that the trailing zero is significant; there are three significant figures in this value.

6. Trailing zeros in a whole number with no decimal shown are not significant: Writing just "540" indicates that the trailing zero is not significant, and therefore, there are only two significant figures in this value.

7. Exact numbers have an infinite number of significant figures: This rule applies to numbers that are definitions. For example, $$\pu{1 meter} = \pu{1.00 meters} = \pu{1.0000 meters} = \pu{1.0000000000000000000 meters}$$, etc.

8. For a number in scientific notation: Consider a number $$N \times 10^x$$. All digits comprising $$N$$ are significant by the first 6 rules; "$$10$$" and "$$x$$" are not significant. For example, $$5.02 \times 10^4$$ has three significant figures: "$$5.02$$." "$$10$$" and "$$4$$" are not significant.

Rule 8 provides the opportunity to change the number of significant figures in a value by manipulating its form. For example, let's try writing 1100 with three significant figures. By rule 6, 1100 has two significant figures; its two trailing zeros are not significant. If we add a decimal to the end, we have 1100., with four significant figures (by rule 5). But by writing it in scientific notation: $$1.10 \times 10^3$$, we create a three-significant-figure value.

By rule 7, $$100$$ in your answer is exact number, hence carries infinite significant figures. Thus, your answer should have been $$0.24$$, with two significant figures (I assume you are familiar with rounding rules).

Now, let's apply Rule 8 to your answer: $$\frac{0.021}{8.925} \times 100 = \frac{0.021}{8.925} \times 10^2 = 0.0024 \times 10^2 = 2.4 \times 10^{-1}$$ This final answer is a number in scientific notation, and hence has two significant figures.

• Thanks alot! perfectly explained! – Sara_jane Jun 15 '19 at 11:23

This is the chemistry part of stackexchange but I can attempt to help you with your question regardless.

$$\frac{8.925-8.904}{8.925} \times 100$$

which can be simplified to

$$\frac{0.025}{8.925} \times 100$$ $$= 0.00235294117... \times 100$$ $$= 0.235...$$

which rounds up to $$0.24$$.

Multiplying by $$100$$ moves the decimal point two places to the right, one for each $$0$$ in $$100$$.

This is a 2 sig.fig answer as the first zero does not count towards significant figures. They only count from your first non-zero digit from left to right.

I hope that helps! (This is my first answer by the way too!)

• why is it a 2 sig fig answer and not 1? why did we ignore that 100 is a 1 sig fig number – Sara_jane Jun 12 '19 at 20:56
• is it because its an exact number so we ignore it? – Sara_jane Jun 12 '19 at 20:57
• You can't answer that question without more context. – Zhe Jun 12 '19 at 21:00
• The equation looks like a formula to calculate a percentage. If so, then yes, the 100 is an "exact number." – MaxW Jun 12 '19 at 22:26