QuanticaPrealgebraOpen in the coursePercent means per hundred, from the Latin per centum. So \(73\%\) is literally \(\frac{73}{100}\), a fraction whose denominator is \(100\). The \(\%\) sign is just a stand-in for that \(/100\).
Problem
Your phone storage is \(80\%\) full. Write \(80\%\) as a fraction in lowest terms.
Show a hint
- The word percent means "per hundred," so put the number over 100. That gives you \(\tfrac{80}{100}\), and now it is just a fraction waiting to be simplified.
- Both 80 and 100 are divisible by 20. Divide the top by 20 and the bottom by 20 to land on the smallest equivalent fraction.
Show the full solution
\(80\%\) means 80 per hundred, so \(80\% = \frac{80}{100}\). The greatest common factor of 80 and 100 is 20, so divide the top and the bottom by 20. $$\frac{80}{100} = \frac{80 \div 20}{100 \div 20} = \frac{4}{5}$$ So the answer is \(\boxed{4/5}\). Reducing does not shrink the amount, it just names the same slice of storage with smaller numbers.
Problem
A bakery sold \(250\%\) as many sourdough loaves as last Saturday. Percents can exceed \(100\). Write \(250\%\) as a fraction in lowest terms.
Show a hint
- A percent means that many hundredths, so \(250\%\) is the same as \(\tfrac{250}{100}\). Going above \(100\) is allowed, it simply gives a fraction larger than \(1\).
- Now reduce \(\tfrac{250}{100}\). Both the top and bottom share a factor of \(50\), so divide each by \(50\).
Show the full solution
A percent is a number of hundredths, so \(250\% = \frac{250}{100}\). The greatest common factor of \(250\) and \(100\) is \(50\), so divide the top and the bottom by \(50\). $$\frac{250}{100} = \frac{250 \div 50}{100 \div 50} = \frac{5}{2}$$ So \(250\%\) in lowest terms is \(\boxed{5/2}\). Going past \(100\) is fine, it just means more than one whole, and \(\tfrac{5}{2}\) is \(2.5\), the two and a half times the bakery sold.
Problem
A glacier retreated by \(-45\%\) over a decade. A percent can be negative. Write \(-45\%\) as a fraction in lowest terms.
Show a hint
- Start from the meaning. \(-45\%\) means \(-45\) out of \(100\), so write it as \(\tfrac{-45}{100}\). The negative sign just rides along on top.
- Now reduce \(\tfrac{-45}{100}\). Both \(45\) and \(100\) share a factor of \(5\), so divide each by \(5\) and keep the minus sign.
Show the full solution
\(-45\%\) means \(-45\) out of \(100\), so \(-45\% = \tfrac{-45}{100}\). Both \(45\) and \(100\) are divisible by \(5\), and dividing gives \(\tfrac{-45 \div 5}{100 \div 5} = \tfrac{-9}{20}\). Since \(9\) and \(20\) share no factor bigger than \(1\), that is lowest terms, so \(\boxed{-9/20}\). The minus sign rides along on the numerator and changes nothing about how you reduce.
Problem
A coupon takes \(38\%\) off. Rewrite \(38\%\) as a decimal.
Show a hint
- A percent is a count per hundred, so \(38\%=\tfrac{38}{100}=38\div 100\). You are not taking the percent of any price yet, just rewriting the number itself.
- Dividing by \(100\) slides the decimal point two places to the left. Picture \(38\) as \(38.\) and hop that point left twice, filling the empty spot with a zero.
Show the full solution
\(38\%=\dfrac{38}{100}=38\div 100\), and dividing by \(100\) moves the decimal point two places to the left. Start at \(38.\), hop the point left once to get \(3.8\), then left again to land at \(0.38\). So \(38\%=\boxed{0.38}\).
Problem
A bridge sensor drifts by \(0.4\%\). Write \(0.4\%\) as a decimal.
Show a hint
- A percent means "divided by one hundred", so dividing by \(100\) slides the decimal point two places to the left. Start from \(0.4\) and move the point.
- Moving the point in \(0.4\) two places left runs out of digits, so fill the empty spots with zeros. You get \(0.4 \to 0.04 \to 0.004\).
Show the full solution
A percent is a count out of one hundred, so \(0.4\%\) means \(\tfrac{0.4}{100}\). Dividing by \(100\) slides the decimal point two places to the left. Starting at \(0.4\), one slide gives \(0.04\) and a second slide gives \(0.004\), with zeros filling the empty places. So \(0.4\% = \boxed{0.004}\).
Problem
Your fitness app shows daily goal progress as the decimal \(0.41\). Write \(0.41\) as a percent. Give just the number.
Show a hint
- A percent is a count out of one hundred, so to name a decimal as a percent you multiply it by \(100\). Multiplying by \(100\) slides the decimal point two places to the right.
- Slide the point in \(0.41\) two places right. It moves past the \(4\) and past the \(1\), landing after the \(1\), which leaves \(41\).
Show the full solution
A decimal becomes a percent when you multiply it by \(100\), and multiplying by \(100\) just slides the decimal point two places to the right. Start with \(0.41\). Slide the point past the \(4\) to get \(4.1\), then past the \(1\) to get \(41\). So \(0.41 \times 100 = 41\), which means \(0.41 = 41\%\). The answer is \(\boxed{41}\).
Problem
A race lap timer reads \(0.073\) of the way through the final lap. Write \(0.073\) as a percent. Give just the number.
Show a hint
- To go from a decimal to a percent, multiply by \(100\), which slides the decimal point two places to the right. Start by moving it from \(0.073\) one place, then a second place.
- Sliding \(0.073\) two places right gives \(007.3\). Drop the leading zeros and you are left with a number that still has a digit after the point.
Show the full solution
To make a decimal into a percent, multiply by \(100\), which slides the decimal point two places to the right. From \(0.073\), one slide gives \(0.73\) and a second gives \(7.3\), so \(0.073 = 7.3\%\) and the number you type is \(\boxed{7.3}\). There were three digits after the point, so only two of them clear it and the last one stays behind in the tenths place. That is why the percent is itself a decimal.
Problem
A garden bed is \(\tfrac{3}{4}\) planted. A percent has a hidden denominator of \(100\). Write \(\tfrac{3}{4}\) as a percent. Give just the number.
Show a hint
- You want the same fraction with a denominator of \(100\). Ask yourself what \(4\) must be multiplied by to reach \(100\), and remember to multiply the top by that same number.
- Since \(4 \times 25 = 100\), multiply top and bottom by \(25\). That gives \(\tfrac{3 \times 25}{4 \times 25} = \tfrac{75}{100}\), and the numerator over \(100\) is the percent.
Show the full solution
Rewrite \(\tfrac{3}{4}\) as an equal fraction whose denominator is \(100\). Since \(4 \times 25 = 100\), multiply the top and bottom by the same \(25\). $$\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100}.$$ A fraction over \(100\) is read directly as a percent, so \(\tfrac{75}{100} = 75\%\). The percent is \(\boxed{75}\).
Problem
A bakery scales a recipe to \(\tfrac{8}{5}\) of the original. Since \(\tfrac{8}{5} > 1\), the percent lands above \(100\). Write \(\tfrac{8}{5}\) as a percent. Answer with just the number.
Show a hint
- To get a denominator of \(100\), ask what turns \(5\) into \(100\). Since \(5\times 20=100\), multiply both the top and the bottom by \(20\).
- Multiplying gives \(\tfrac{8\times 20}{5\times 20}=\tfrac{160}{100}\). The numerator over \(100\) is the percent.
Show the full solution
Since \(5\times 20=100\), rewrite \(\tfrac{8}{5}\) with a denominator of \(100\) by multiplying the top and bottom by \(20\). $$\frac{8}{5}=\frac{8\times 20}{5\times 20}=\frac{160}{100}$$ A fraction over \(100\) reads straight off as a percent, so the answer is \(\boxed{160}\). Because \(\tfrac{8}{5}\) is bigger than one whole, the percent lands above \(100\), exactly as expected.
Problem
A recipe calls for \(\tfrac{9}{8}\) cups of oats. Since eighths do not divide \(100\) evenly, multiply by \(100\) and simplify. Write \(\tfrac{9}{8}\) as a percent. Give just the number.
Show a hint
- To turn a fraction into a percent, multiply it by \(100\). So you want \(\tfrac{9}{8}\times 100\), which is the same as \(\tfrac{900}{8}\).
- Now simplify \(\tfrac{900}{8}\) by dividing top and bottom by \(4\) to get \(\tfrac{225}{2}\), then divide \(225\) by \(2\). It does not come out even, so expect a decimal ending in a half.
Show the full solution
A fraction becomes a percent when you multiply it by \(100\). $$\frac{9}{8}\times 100=\frac{9\times 100}{8}=\frac{900}{8}$$ Divide the top and bottom by \(4\) to get \(\tfrac{225}{2}\), and \(225\div 2 = 112.5\). So the number is \(\boxed{112.5}\). It clears \(100\) because \(\tfrac{9}{8}\) is more than one whole, and it carries a \(.5\) because halving an odd number leaves a half.
Problem
Room A chose pizza at \(65\%\). Room B chose pizza at \(\tfrac{3}{5}\). Convert both to decimals and pick the larger share. Type \(1\) for Room A or \(2\) for Room B.
Show a hint
- You cannot compare a percent and a fraction while they look different. Turn both into decimals first. For the percent, slide the point two places left. For the fraction, divide the top by the bottom.
- Room A is \(65\% = 0.65\). Room B is \(\tfrac{3}{5} = 0.60\). Now both are decimals, so just compare \(0.65\) with \(0.60\) and name the option that is bigger.
Show the full solution
Put both shares in decimal form. Room A is \(65\%\), and dividing by \(100\) slides the point two places left, so \(65\% = 0.65\). Room B is \(\tfrac{3}{5}\), and \(3 \div 5 = 0.60\), which you can check with \(\tfrac{3}{5} = \tfrac{6}{10} = 0.60\). Since \(0.65 > 0.60\), Room A has the larger share, option \(\boxed{1}\). Values in different forms cannot be judged by eye, so convert them to one shared form first.
Problem
A sales tax is \(12.5\%\). Write \(12.5\%\) as a fraction in lowest terms.
Show a hint
- Percent means per hundred, so start by writing \(12.5\%\) as \(\tfrac{12.5}{100}\). A fraction in lowest terms cannot keep a decimal upstairs, so you need to clear that half.
- Multiply the top and bottom by \(10\) to get rid of the decimal point. That turns \(\tfrac{12.5}{100}\) into \(\tfrac{125}{1000}\), and now both numbers are whole and ready to simplify.
Show the full solution
Percent means per hundred, so \(12.5\% = \tfrac{12.5}{100}\). Multiply the top and bottom by \(10\) to clear the decimal. $$\frac{12.5}{100} = \frac{12.5 \times 10}{100 \times 10} = \frac{125}{1000}$$ Both \(125\) and \(1{,}000\) share the factor \(125\), and dividing each by it gives \(\boxed{1/8}\). Twelve and a half percent being exactly one eighth makes sense, since \(\tfrac{1}{8}\) of a hundred is \(12.5\).
Practice these ideas
Practice
A survey found \(55\%\) of students bike to school. Write \(55\%\) as a fraction in lowest terms.
Show the solution
Percent means "per hundred," so \(55\%\) is \(55\) out of \(100\). $$55\% = \frac{55}{100}$$ Now reduce. The numerator and denominator both share a factor of \(5\), so divide each by \(5\). $$\frac{55}{100} = \frac{55 \div 5}{100 \div 5} = \frac{11}{20}$$ Since \(11\) is prime and does not divide \(20\), the fraction is in lowest terms. \(\boxed{11/20}\)
Practice
A pollster reports \(24\%\) of people surveyed had never tasted dark chocolate. Write \(24\%\) as a fraction in lowest terms.
Show the solution
The word percent means per hundred, so \(24\%\) is \(24\) out of \(100\). $$24\% = \frac{24}{100}$$ Now simplify. The greatest factor shared by \(24\) and \(100\) is \(4\), so divide both the numerator and the denominator by \(4\). $$\frac{24}{100} = \frac{24 \div 4}{100 \div 4} = \frac{6}{25}$$ Since \(6\) and \(25\) share no common factor larger than \(1\), this is fully reduced. \(\boxed{6/25}\)
Practice
A loyalty card stamps your total at \(320\%\) of last month. Write \(320\%\) as a fraction in lowest terms.
Show the solution
Percent means per hundred, so \(320\%=\tfrac{320}{100}\). The greatest common factor of \(320\) and \(100\) is \(20\), so divide both parts by it. That gives \(\tfrac{320\div 20}{100\div 20}=\tfrac{16}{5}\). Since \(16\) and \(5\) share no factor bigger than \(1\), this is lowest terms, and it is more than one whole because \(16>5\). \(\boxed{16/5}\)
Practice
A deep freezer drifted \(-8\%\) from its set point. Write \(-8\%\) as a fraction in lowest terms.
Show the solution
Percent means per hundred, so the number sits over \(100\) and keeps its sign. That gives \(-8\% = \tfrac{-8}{100}\). The top and bottom share a common factor of \(4\), so divide both by \(4\). The numerator becomes \(-8 \div 4 = -2\) and the denominator becomes \(100 \div 4 = 25\). Since \(2\) and \(25\) have no common factor left, the fraction is in lowest terms. So \(-8\%\) equals \(\boxed{-2/25}\).
Practice
A weather app says the chance of rain is \(73\%\). Write \(73\%\) as a decimal.
Show the solution
A percent is just a number of hundredths, so \(73\%\) means \(\tfrac{73}{100}\). Dividing by \(100\) slides the decimal point two places to the left, turning \(73.\) into \(0.73\). So the decimal form is \(\boxed{0.73}\).
Practice
Write \(108\%\) as a decimal.
Show the solution
To write a percent as a decimal you divide by \(100\), which slides the point two places to the left. Begin with \(108\), really \(108.0\), and move the point two places left to land between the \(1\) and the first \(0\). So \(108\% = \tfrac{108}{100} = \boxed{1.08}\). The result is more than \(1\) because \(108\%\) is more than a whole.
Practice
A lab label reads \(0.6\%\) saline. Write \(0.6\%\) as a decimal.
Show the solution
A percent is just a number over 100, so \(0.6\%\) means \(\tfrac{0.6}{100}\). Dividing by 100 slides the decimal point two places to the left. Starting from \(0.6\), one place left gives \(0.06\), and a second place gives \(0.006\), with placeholder zeros filling the gaps. So \(0.6\%\) as a decimal is \(\boxed{0.006}\).
Practice
A probe is filled to \(0.47\) of its tank. Write \(0.47\) as a percent. Give just the number.
Show the solution
To turn a decimal into a percent, multiply by \(100\), which slides the point two places to the right. So \(0.47 \times 100 = 47\), and the number to post is \(\boxed{47}\). It also reads straight off, since \(0.47 = \tfrac{47}{100}\) is already \(47\) per hundred.
Practice
A grain of gold dust registers \(0.025\) of a carat. Write \(0.025\) as a percent. Give just the number.
Show the solution
To go from a decimal to a percent, multiply by \(100\), which slides the decimal point two places to the right. $$0.025 \times 100 = 2.5$$ Sliding the point from \(0.025\) two places to the right lands between the \(2\) and the \(5\), so the value is \(2.5\%\). The number alone is \(\boxed{2.5}\).
Practice
A trail map marks \(\tfrac{7}{20}\) of its loops as beginner-friendly. Write \(\tfrac{7}{20}\) as a percent. Give just the number.
Show the solution
Multiplying \(20\) by \(5\) gives \(100\), so multiply the top by \(5\) as well to keep the same value. $$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100}$$ A fraction over \(100\) reads straight off as a percent, so the number is \(\boxed{35}\). Rescaling like this works whenever the denominator divides \(100\) evenly.
Practice
A juice bottle is \(\tfrac{5}{8}\) real fruit. Write \(\tfrac{5}{8}\) as a percent. Give just the number.
Show the solution
To turn a fraction into a percent, multiply it by \(100\). $$\frac{5}{8}\times 100=\frac{500}{8}=62.5$$ So \(\tfrac{5}{8}\) of the bottle is \(62.5\%\) fruit, and the number is \(\boxed{62.5}\).
Practice
Write \(\tfrac{11}{5}\) as a percent. Give just the number.
Show the solution
To write a fraction as a percent, multiply it by \(100\). That gives \(\tfrac{11}{5}\times 100=\tfrac{1{,}100}{5}\), and \(1{,}100\div 5=220\), so \(\tfrac{11}{5}=220\%\) and the number is \(\boxed{220}\). That checks out, since \(\tfrac{11}{5}\) is a bit more than \(2\) and \(220\%\) is a bit more than \(200\%\).
Practice
A pool is \(87.5\%\) full. Write \(87.5\%\) as a fraction in lowest terms.
Show the solution
Percent means per hundred, so \(87.5\%=\tfrac{87.5}{100}\). Multiply the top and bottom by \(10\) to clear the decimal, which gives \(\tfrac{875}{1000}\). Now reduce. Both numbers are divisible by \(125\), and \(875\div125=7\) while \(1{,}000\div125=8\), so the fraction becomes \(\tfrac{7}{8}\). Since \(7\) is prime and does not divide \(8\), this is lowest terms. \(\boxed{7/8}\)
Practice
Option 1 is \(\tfrac{9}{20}\) and option 2 is \(44\%\). Which value is larger? Type \(1\) or \(2\).
Show the solution
Put both into percent form so they can be compared head to head. For the fraction, scale \(\tfrac{9}{20}\) to a denominator of \(100\) by multiplying top and bottom by \(5\), which gives \(\tfrac{45}{100}=45\%\). Now line them up, \(45\%\) against \(44\%\). Since \(45\%>44\%\), option 1 is the larger value. \(\boxed{1}\)