History of Women Inventors Who Filed Patents

History of Women Inventors Who Filed Patents Before the 1970s, the topic of women in history was largely missing from general public consciousness. To address this situation, the Education Task Force on the Status of Women initiated a Womens History Week celebration in 1978 and chose the week of March 8 to coincide with International Womens Day. In 1987, the National Womens History Project petitioned Congress to expand the celebration to the entire month of March. Since then, the National Womens History Month Resolution has been approved every year with bipartisan support in both the House and Senate. The First Woman to File an American Patent In 1809, Mary Dixon Kies received the first U. S. patent issued to a woman. Kies, a Connecticut native, invented a process for weaving straw with silk or thread. First Lady Dolley Madison praised her for boosting the nations hat industry. Unfortunately, the patent file was destroyed in the great Patent Office fire in 1836. Until about 1840, only 20 other patents were issued to women. The inventions related to apparel, tools, cook stoves, and fireplaces. Naval Inventions In 1845, Sarah Mather received a patent for the invention of a submarine telescope and lamp. This was a remarkable device that permitted sea-going vessels to survey the depths of the ocean. Martha Coston perfected then patented her deceased husbands idea for a pyrotechnic flare. Costons husband, a former naval scientist, died leaving behind only a rough sketch in a diary of plans for the flares. Martha developed the idea into an elaborate system of flares called Night Signals that allowed ships to communicate messages nocturnally. The U. S. Navy bought the patent rights to the flares. Costons flares served as the basis of a system of communication that helped to save lives and to win battles. Martha credited her late husband with the first patent for the flares, but in 1871 she received a patent for an improvement exclusively her own. Paper Bags Margaret Knight was born in 1838. She received her first patent at the age of 30, but inventing was always part of her life. Margaret or Mattie as she was called in her childhood, made sleds and kites for her brothers while growing up in Maine. When she was just 12 years old, she had an idea for a stop-motion device that could be used in textile mills to shut down machinery, preventing workers from being injured. Knight eventually received some 26 patents. Her machine that made flat-bottomed paper bags is still used to this very day! 1876 Philadelphia Centennial Exposition The 1876 Philadelphia Centennial Exposition was a World Fair-like event held to celebrate the amazing progress of the century-old United States of America. The leaders of early feminist and womens suffrage movements had to aggressively lobby for the inclusion of a womans department in the exposition. After some firm pressing, the Centennial Womens Executive Committee was established, and a separate Womans Pavilion erected. Scores of women inventors either with patents or with patents pending displayed their inventions. Among them was Mary Potts and her invention Mrs. Potts Cold Handle Sad Iron patented in 1870. Chicagos Columbian Exposition in 1893 also included a Womans Building. A unique safety elevator invented by multi-patent holder Harriet Tracy and a device for lifting and transporting invalids invented by Sarah Sands were among the many items featured at this event. Traditionally womens undergarments consisted of brutally tight corsets meant to shape womens waists into unnaturally small forms. Some suggested that the reason women seemed so fragile, expected to faint at any time, was because their corsets prohibited proper breathing. Enlightened womens groups throughout the nation resoundingly agreed that less restrictive underclothing was in order. Susan Taylor Converses one-piece flannel Emancipation Suit, patented August 3, 1875, eliminated the need for a suffocating corset and became an immediate success. A number of womens groups lobbied for Converse to give up the 25-cent royalty she received on each Emancipation Suit sold, an effort that she rejected. Linking the emancipation of women from constrictive undergarments to her own freedom to profit from her intellectual property, Converse responded: With all your zeal for womens rights, how could you even suggest that one woman like myself should give of her head and hand labor without fair compensation? Perhaps its a no-brainer that women inventors should turn their minds to making better the things that often concern women the most. The Ultimate Home The ultimate convenience invention must certainly be woman inventor Frances Gabe’s self-cleaning house. The house, a combination of some 68 time-, labor-, and space-saving mechanisms, makes the concept of housework obsolete. Each of the rooms in the termite-proof, cinder block constructed, the self-cleaning house is fitted with a 10-inch, ceiling-mounted cleaning/drying/heating/cooling device. The walls, ceilings, and floors of the house are covered with resin, a liquid that becomes water-proof when hardened. The furniture is made of a water-proof composition, and there are no dust-collecting carpets anywhere in the house. At the push of a sequence of buttons, jets of soapy water wash the entire room. Then, after a rinse, the blower dries up any remaining water that hasn’t run down the sloping floors into a waiting drain. The sink, shower, toilet, and bathtub all clean themselves. The bookshelves dust themselves while a drain in the fireplace carries away ashes. The clothes closet is also a washer/drier combination. The kitchen cabinet is also a dishwasher; simply pile in soiled dishes, and don’t bother taking them out until they are needed again. Not only is the house of practical appeal to overworked homeowners, but also to physically handicapped people and the elderly. Frances Gabe (or Frances G. Bateson) was born in 1915 and now resides comfortably in Newberg, Oregon in the prototype of her self-cleaning house. Gabe gained experience in housing design and construction at an early age from working with her architect father. She entered the Girl’s Polytechnic College in Portland, Oregon at age 14 finishing a four-year program in just two years. After World War II, Gabe with her electrical engineer husband started a building repairs business that she ran for more than 45 years. In addition to her building/inventing credits, Frances Gabe is also an accomplished artist, musician, and mother. Fashion Forward Fashion designer Gabriele Knecht realized something that clothes makers were neglecting in their clothing designs- that our arms come out of our sides in a slightly forward direction, and we work them in front of our bodies. Knecht’s patented Forward Sleeve design is based on this observation. It lets the arms move freely without shifting the whole garment and allows clothes to drape gracefully on the body. Knecht was born in Germany in 1938 and came to America when she was 10 years old. She studied fashion design, and in 1960, received a bachelor of fine arts degree from Washington University in St. Louis. Knecht also took courses in physics, cosmology, and other areas of science that may seem unrelated to the fashion industry. Her broadened knowledge, however, helped her understand shapes and methods of pattern design. In 10 years she filled 20 notebooks with sketches, analyzed all the angles that sleeves can take, and made 300 experimental patterns and garments. Although Knecht had been a successful designer for several New York companies, she felt she had more creative potential. Struggling to start her own business, Knecht met a buyer from Saks Fifth Avenue department store who liked Knecht’s designs. Soon she was creating them exclusively for the store, and they sold well. In 1984 Knecht received the first annual More Award for the best new designer of women’s fashions. Carol Wior is the woman inventor of the Slimsuit, a swimsuit guaranteed to take an inch or more off the waist or tummy and to look natural. The secret to a slimmer look in the inner lining that shapes the body in specific areas, hiding bulges and giving a smooth, firm appearance. The Slimsuit comes with a tape measure to prove the claim. Wior was already a successful designer when she envisioned the new swimsuit. While on vacation in Hawaii, she always seemed to be pulling and tugging on her swimsuit to try to get it to cover properly, all the while trying to hold in her stomach. She realized other women were just as uncomfortable and began to think of ways to make a better swimsuit. Two years and a hundred trail patterns later, Wior achieved the design she wanted. Wior began her designing career at only 22 years old in her parents garage in Arcadia, California. With $77 and three sewing machines bought at auction, she made classic, elegant but affordable dresses and delivered them to her customers in an old milk truck. Soon she was selling to major retail stores and was quickly building a multi-million dollar business. At age 23, she was one of the youngest fashion entrepreneurs in Los Angeles. Protecting the Children When Ann Moore was a Peace Corps volunteer, she observed mothers in French West Africa carrying their babies securely on their backs. She admired the bonding between the African mother and child and wanted the same closeness when she returned home and had her own baby. Moore and her mother designed a carrier for Moores daughter similar to those she saw in Togo. Ann Moore and her husband formed a company to make and market the carrier, called the Snugli (patented in 1969). Today babies all over the world are being carried close to their mothers and fathers. In 1912, the beautiful soprano opera singer and actress of the late 19th and early 20th centuries, Lillian Russell, patented a combination dresser-trunk built solidly enough to remain intact during travel and doubled as a portable dressing room. Silver Screen superstar Hedy Lamarr (Hedwig Kiesler Markey) with the help of composer George Antheil invented a secret communication system in an effort to help the allies defeat the Germans in World War II. The invention, patented in 1941, manipulated radio frequencies between transmission and reception to develop an unbreakable code so that top-secret messages could not be intercepted. Julie Newmar, a living Hollywood film and television legend, is a women inventor. The former Catwoman patented ultra-sheer, ultra-snug pantyhose. Known for her work in films such as Seven Brides for Seven Brothers and Slaves of Babylon, Newmar has also appeared recently in Fox Televisions Melrose Place and the hit feature-film To Wong Fu, Thanks for Everything, Love Julie Newmar. Ruffles, fluted collars, and pleats were very popular in Victorian-era clothing. Susan Knoxs fluting iron made pressing the embellishments easier. The trademark featured the inventors picture and appeared on each iron. Women have made many contributions to advance the fields of science and engineering. Nobel Prize Winner Katherine Blodgett (1898-1979) was a woman of many firsts. She was the first female scientist hired by General Electric’s Research Laboratory in Schenectady, New York (1917) as well as the first woman to earn a Ph.D. in Physics from Cambridge University (1926). Blodgett’s research on monomolecular coatings with Nobel Prize-winning Dr. Irving Langmuir led her to a revolutionary discovery. She discovered a way to apply the coatings layer by layer to glass and metal. The thin films, which naturally reduced glare on reflective surfaces, when layered to a certain thickness, would completely cancel out the reflection from the surface underneath. This resulted in the world’s first 100% transparent or invisible glass. Blodgett’s patented film and process (1938) has been used for many purposes including limiting distortion in eyeglasses, microscopes, telescopes, camera, and projector lenses. Programming Computers Grace Hopper (1906-1992) was one of the first programmers to transform large digital computers from oversized calculators into relatively intelligent machines capable of understanding human instructions. Hopper developed a common language with which computers could communicate called Common Business-Oriented Language or COBOL, now the most widely used computer business language in the world. In addition to many other firsts, Hopper was the first woman to graduate from Yale University with a Ph.D. in Mathematics, and in 1985, was the first woman ever to reach the rank of admiral in the US Navy. Hopper’s work was never patented; her contributions were made before computer software technology was even considered a patentable field. Invention of Kevlar Stephanie Louise Kwolek’s research with high-performance chemical compounds for the DuPont Company led to the development of a synthetic material called Kevlar which is five times stronger than the same weight of steel. Kevlar, patented by Kwolek in 1966, does not rust nor corrode and is extremely lightweight. Many police officers owe their lives to Stephanie Kwolek, for Kevlar is the material used in bulletproof vests. Other applications of the compound include underwater cables, brake linings, space vehicles, boats, parachutes, skis, and building materials. Kwolek was born in New Kensington, Pennsylvania in 1923. Upon graduating in 1946 from the Carnegie Institute of Technology (now Carnegie-Mellon University) with a bachelor’s degree, Kwolek went to work as a chemist at the DuPont Company. She would ultimately obtain 28 patents during her 40-year tenure as a research scientist. In 1995, Kwolek was inducted into the Hall of Fame. Inventors NASA Valerie Thomas received a patent in 1980 for inventing an illusion transmitter. This futuristic invention extends the idea of television, with its images located flatly behind a screen, to having three-dimensional projections appear as though they were right in your living room. Perhaps in the not-so-distant future, the illusion transmitter will be as popular as the TV is today. Thomas worked as a mathematical data analyst for NASA after receiving a degree in physics. She later served as project manager for the development of NASA’s image-processing system on Landsat, the first satellite to send images from outer space. In addition to having worked on several other high-profile NASA projects, Thomas continues to be an outspoken advocate for minority rights. Barbara Askins, a former teacher, and mother, who waited until after her two children entered school to complete her B. S. in chemistry followed by a Master’s degree in the same field, developed a totally new way of processing film. Askins was hired in 1975 by NASA to find a better way to develop astronomical and geological pictures taken by researchers. Until Askins’ discovery, these images, while containing valuable information, were hardly visible. In 1978 Askins patented a method of enhancing the pictures using radioactive materials. The process was so successful that its uses were expanded beyond NASA research to improvements in X-ray technology and in the restoration of old pictures. Barbara Askins was named National Inventor of the Year in 1978. Ellen Ochoa’s pre-doctoral work at Stanford University in electrical engineering led to the development of an optical system designed to detect imperfections in repeating patterns. This invention, patented in 1987, can be used for quality control in the manufacturing of various intricate parts. Dr. Ochoa later patented an optical system which can be used to robotically manufacture goods or in robotic guiding systems. In all Ellen Ochoa has received three patents, most recently in 1990. In addition to being a woman inventor, Dr. Ochoa is also a research scientist and astronaut for NASA who has logged hundreds of hours in space. Inventing Geobond Patricia Billings received a patent in 1997 for a fire resistant building material called Geobond. Billings’ work as a sculpture artist put her on a journey to find or develop a durable additive to prevent her painstaking plaster works from accidentally falling and shattering. After nearly two decades of basement experiments, the result of her efforts was a solution which when added to a mixture of gypsum and concrete, creates an amazingly fire resistant, indestructible plaster. Not only can Geobond add longevity to artistic works of plastic, but also it is steadily being embraced by the construction industry as an almost universal building material. Geobond is made with non-toxic ingredients which make it the ideal replacement for asbestos. Currently, Geobond is being sold in more than 20 markets worldwide, and Patricia Billings, great grandmother, artist, and woman inventor remains at the helm of her carefully constructed Kansas City-based empire. Women care and women care as inventors. Many female inventors have turned their skills on finding ways to save lives. Invention of Nystatin As researchers for the New York Department of Health, Elizabeth Lee Hazen and Rachel Brown combined their efforts to develop the anti-fungal antibiotic drug Nystatin. The drug, patented in 1957 was used to cure many disfiguring, disabling fungal infections as well as to balance the effect of many antibacterial drugs. In addition to human ailments, the drug has been used to treat such problems as Dutch Elms disease and to restore water-damaged artwork from the effects of mold. The two scientists donated the royalties from their invention, over $13 million dollars, to the nonprofit Research Corporation for the advancement of academic scientific study. Hazen and Brown were inducted into the National Inventors Hall of Fame in 1994. Fighting Disease Gertrude Elion patented the leukemia-fighting drug 6-mercaptopurine in 1954 and has made a number of significant contributions to the medical field. Dr. Elions research led to the development of Imuran, a drug that aids the body in accepting transplanted organs, and Zovirax, a drug used to fight herpes. Including 6-mercaptopurine, Elions name is attached to some 45 patents. In 1988 she was awarded the Nobel Prize in Medicine with George Hitchings and Sir James Black. In retirement, Dr. Elion, who was inducted into the Hall of Fame in 1991, continues to be an advocate for medical and scientific advancement. Stem Cell Research Ann Tsukamoto is co-patenter of a process to isolate the human stem cell; the patent for this process was awarded in 1991. Stem cells are located in bone marrow and serve as the foundation for the growth of red and white blood cells. Understanding how stem cells grow or how they might be artificially reproduced is vital to cancer research. Tsukamotos work has led to great advancements in comprehending the blood systems of cancer patients and may one day lead to a cure for the disease. She is currently directing further research in the areas of stem cell growth and cellular biology. Patient Comfort Betty Rozier and Lisa Vallino, a mother and daughter team, invented an intravenous catheter shield to make the use of IVs in hospitals safer and easier. The computer-mouse shaped, polyethylene shield covers the site on a patient where an intravenous needle has been inserted. The IV House prevents the needle from being accidentally dislodged and minimizes its exposure to patient tampering. Rozier and Vallino received their patent in 1993. After fighting breast cancer and undergoing a mastectomy in 1970, Ruth Handler, one of the creators of the Barbie Doll, surveyed the market for a suitable prosthetic breast. Disappointed in the options available, she set about designing a replacement breast that was more similar to a natural one. In 1975, Handler received a patent for Nearly Me, a prosthesis made of material close in weight and density to natural breasts.

Complete Guide to Fractions and Ratios on SAT Math

Complete Guide to Fractions and Ratios on SAT Math SAT / ACT Prep Online Guides and Tips You likely had your first taste of working with fractions sometime in elementary school, though it's probably been a while since you've had to deal with how they shift, change, and interact with one another. To refresh, fractions and ratios are both used to represent pieces of a whole. Fractions tell you how many pieces you have compared to a potential whole amount (3 red marbles in a bag of 5, for example), while ratios compare pieces to each other (3 red marbles to 2 blue marbles) or, more rarely, pieces to the whole amount (again, 3 red marbles in 5 total). If this sounds complicated to you right now, don’t worry! We will go through all the principles behind fractions and ratios in this guide. If this seems easy to you right now, definitely check out the practice problems at the end of the guide to make sure you have mastered all the different kinds of fraction and ratio problems you’ll see on the test. The SAT likes to present familiar concepts in unfamiliar ways, so don’t let your mastery of fractions lead you to make assumptions about how you’ll see fractions and ratios on the test. No matter how comfortable you are (or are not) with fractions and ratios right now, this guide is for you. Here, we will go through the complete breakdown of fractions and ratios on the SAT- what they mean, how to manipulate them, and how to answer the most difficult fraction and ratio problems on the SAT. This Guide This guide is seperated into two distinct categories- everything you need to know about fractions and everything you need to know about ratios. For each section, we will go through the ins and outs of what fractions and ratios mean as well as how to manipulate and solve the different kinds of fraction and ratio problems you'll see on the SAT. We will also breakdown how you can tell when an SAT problem requires a ratio or a fraction and how to set up your approach these kinds of problems. At the end, you will be able to test your knowledge on real SAT math questions. The more you prep for the SAT, the more your brain can be Swiss-army-knife-ready for any question the test can throw at you. What are Fractions? $${\a \piece}/{\the \whole}$$ Fractions are pieces of a whole. They are expressed as the amount you have (the numerator) over the whole (the denominator). A pizza is divided into 8 pieces. Kyle ate 3 pieces. What fraction of the pizza did he eat? He ate $3/8$ths of the pizza. 3 is the numerator (top number) because he ate that many pieces of the whole, and 8 is the denominator (bottom number) because there are 8 pieces total (the whole). Math is always more fun when it's delicious. Special Fractions A number over itself equals 1 $3/3=1$ $10/10=1$ $(a+b)/(a+b)=1$ A whole number can be expressed as itself over 1 $5=5/1$ $22/1=22$ $(a+b)/1=a+b$ 0 divided by any number is 0 $0/17=0$ $0/(a+b)=0$ There is one exception to this rule: $0/0=\undefined$. The reason for this lies in the next rule. Any number divided by 0 is undefined Zero cannot act as a denominator. For more information on this check out our guide to advanced integers. But for now all that matters is that you know that 0 cannot act as a denominator. Reducing Fractions If both the numerator and the denominator have a common factor (a number they can both be divided by), then the fraction can be reduced. For the purposes of the SAT, you will need to reduce your fractions to get to your final answer. To reduce a fraction, you must divide both the numerator and the denominator by the same amount. This keeps the fraction consistent and maintains the proper relationship between numerator and denominator. If your fraction is $3/12$, then it can be written as $1/4$. Why? Because both 3 and 12 are divisible by 3. $3/3=1$ and $12/3=4$. So your final fraction is $1/4$ Now let's figure out how to perform the four basic math functions on fractions. Adding or Subtracting Fractions You can add or subtract fractions as long as their denominators are the same. To do so, you keep the denominator consistent and simply add the numerators. $4/15+2/15=6/15$ But you CANNOT add or subtract fractions if your denominators are unequal. $4/15+2/5=?$ So what can you do when your denominators are unequal? You must make them equal by finding a common multiple (number they can both multiply evenly into) of their denominators. In the case of $4/15+2/5$, a common multiple of the denominators 15 5 is 15. When you find a common multiple of the denominators, you must multiply both the numerator and the denominator by the amount it took to achieve that number. Again, this keeps the fraction (the relationship between numerator and denominator) consistent. Think of it as the opposite of reducing a fraction. To get to the common denominator of 15, $4/15$ must be multiplied by $1/1$ Why? Because 15*1=15. $(4/15)(1/1)=4/15$. The fraction remains unchanged. To get to the common denominator of 15, $2/5$ must be multiplied by $3/3$. Why? Because 5*3=15. $(2/5)(3/5)=6/15$. Now we can add them, as they have the same denominator. $4/15+6/15=10/15$ We can further reduce $10/15$ into $2/3$ because both 10 and 15 are divisible by 5. So our final answer is $2/3$. Multiplying Fractions Multiplying fractions is a bit simpler than adding or dividing fractions. There is no need to find a common denominator- you can just multiply the fractions straight across. To multiply a fraction, first multiply the numerators. This product becomes your new numerator. Next, multiply your two denominators. This product becomes your new denominator. $1/4*2/3=(1*2)/(4*3)=2/12$ And again, we reduce our fraction. Both the numerator and the denominator are divisible by 2, so our final answer becomes: $1/6$ Special note: you can speed up the multiplication and reduction process by finding a common factor of your cross multiples before you multiply. $1/4*2/3$ = $1/2*1/3$. Why? Because both 4 and 2 are divisible by 2, we were able to reduce the cross multiples before we even began. This saved us time in reducing the final fraction at the end. So now we can simply say: $1/2*1/3=1/6$. No need to further reduce- our answer is complete. Take note that reducing cross multiples can only be done when multiplying fractions, never while adding or subtracting them! It is also a completely optional step, so do not feel obligated to reduce your cross multiples- you can simply reduce your fraction at the end. Dividing Fractions In order to divide fractions, we must first take the reciprocal (the reversal) of one of the fractions. Afterwards, we simply multiply the two fractions together. Why do we do this? Because division is the opposite of multiplication, so we must reverse one of the fractions to turn it back into a multiplication question. ${2/3}à ·{3/4}$ = $2/3*4/3$ (we took the reciprocal of $3/4$, which means we flipped the fraction upside down to become $4/3$) $2/3*4/3=8/9$ But what happens if you need to divide a fraction by a whole number? If a cake is cut into thirds and each third is cut into fourths, how many pieces of cake are there? *** We start out with $1/3$ of a cake and we need to divide each third 4 more times. Because 4 is a whole number, it can be written as $4/1$. This means that its reciprocal is $1/4$. $1/3à ·4$ = $1/3*1/4=1/12$ Our denominator (the whole) is 12. This means there will be 12 pieces total in the cake. Decimal Points Because fractions are pieces of a whole, you can also express fractions as either a decimal point or a percentage. To convert a fraction into a decimal, simply divide the numerator by the denominator. (The / symbol also acts as a division sign.) $4/5$ = 4/5 = 0.8 Sometimes it is easier to convert a fraction to a decimal in order to work through a problem. This can save you time and effort trying to figure out how to divide or multiply fractions. If $j/k=32$ and $k=3/2$, what is the value of $1/2j$ ? *** As you can see, there are two ways to approach this problem- using fractions and using decimals. We’ll look at both ways. If you were to use fractions, you would set up the problem as a fraction division problem. $k=3/2$ So $j/k=j/{3/2}$ $j/{3/2}$ = $j*2/3$ (remember, we take the reciprocal when we divide) So our full problem looks like this: $2/3*j=32$ Now we must divide 32 by $2/3$ in order to bring it over to the other side and isolate j. This means we need to take the reciprocal yet again. So ${32}/{2/3}$ = $32*3/2=96/2=48$ $j=48$ Now, for the final step, we must take $1/2$ of j. (Note: to "take $1/2$" is the same thing as multiplying by $1/2$.) $48*{1/2}=48/2=24$ Our final answer is 24. Alternatively, we could save ourselves the headache of using fractions and reciprocals and simply use decimals instead. We know that $k=3/2$. Instead of keeping the fraction, let us convert it into a decimal. $3à ·2=1.5$ So $k=1.5$ $j/k=32$ $j/1.5=32$ When you multiply both sides by 1.5, you get: $j=(32)(1.5)=48$ $j=48$ And ${1/2}j={1/2}(48)=24$ So again, our final answer is 24. Percentages After you convert your fraction to a decimal, you can also turn it into a percentage (if needed). So 0.8 from can also be written as 80%, because 0.8*100=80. A pie chart is a useful way of showing relative sizes of fractions and percentages. This shows just how large a fraction $7/10$ (or 70%) truly is. Mixed Fractions Sometimes you may be given a mixed fraction on the SAT. A mixed fraction is a combination of a whole number and a fraction. For example, 7$3/4$ is a mixed fraction. We have a whole number, 7, and a fraction, $3/4$. You can turn a mixed fraction into an ordinary fraction by multiplying the whole number by the denominator and then adding that product to the numerator. The final answer will be ${\the \new \numerator}/{\the \original \denominator}$. 7$3/4$ (7)(4)=28 28+3=31 So your final answer = $31/4$ You must convert mixed fractions into fractions in order to multiply, divide, add, or subtract them with other fractions. In this problem, we began with 5 gallons of water and we ended with 2$1/3$. We must figure out how many gallons we used. 5−2 $5-2{1/3}$ First, let’s get our mixed fraction into a regular fraction. 2$1/3$ = ${[(2*3)+1]}/3={7/3}$ $5/1-7/3$ Now, we need to give each fraction the same denominator. We'll do this by converting $5/1$ into a new fraction with a denominator of 3. $5/1*3/3=15/3$ Finally, we can find the difference between the amounts. $15/3-7/3=8/3$ So we have used up $8/3$rds of the water. Now let’s count how many times the pail was emptied to use up that $8/3$rds of the total water. If you count the dots, the pail was emptied 8 times (the first dot does not count as a time it was emptied- that is merely our starting point). Because the same amount of water was removed each time, we must divide our emptied water by 8. ${8/3}à ·{8/1}$ = $8/3*1/8$ We can now either reduce the cross-multiples (because this is a multiplication problem), which would give us: $8/3*1/8$ = $1/3*1/1$ $1/3*1/1=1/3$ Or we can multiply through and then reduce afterwards: $8/3*1/8=8/24$ $8/12=1/3$ Either way, our final answer is $1/3$; each trip removed $1/3$ of a gallon of water from the tank. Now that we've broken down all there is to know about SAT fractions, let's take a look at their close cousin- the ratio. This shape is called the "golden ratio" and has been studied for thousands of years. It has applications in geometry, nature, and architecture. What are Ratios? Ratios are used as a way to compare one thing to another (or multiple things to one another). If Leslie has 5 white socks and 2 red socks, the white socks and the red socks have a ratio of 5 to 2. Expressing Ratios Ratios can be written in three different ways: A â€Å'to â€Å'B A:B $A/B$ No matter which way you write them, these are all ratios comparing A to B. Different Types of Ratios Just as a fraction represents a part of something out of a whole (written as: ${\a \part}/{\the \whole}$), a ratio can be expressed as either: aâ€Å'part:a â€Å'different â€Å'part OR aâ€Å'part:theâ€Å' whole Because ratios compare values, they can either compare individual pieces to one another or an individual piece to the whole. If Leslie has only 5 white socks and 2 red socks in a drawer, the ratio of white socks to all the socks in the drawer is 5 to 7. (Why 7? Because there are 5 white and 2 red socks, so together they make 5+2=7 socks total.) Some of the many uses of ratios in action (in this case, the ratios are- a part: a different part). Reducing Ratios Just as fractions can be reduced, so too can ratios. Kyle has a stamp collection. 45 of them have pictures of daisies and 30 of them have pictures of roses. What is the ratio of daisy stamps to rose stamps in his collection? *** Right now, the ratio is $45:30$. But they have a common denominator of 15, so this ratio can be reduced. $45/15=3$ $30/15=2$ So the stamps have a ratio of $3:2$ Increasing Ratios Because you can reduce ratios, you can also do the opposite and increase them. In order to do so, you must multiply each piece of the ratio by the same amount (just as you had to divide by the same amount on each side to reduce the ratio). So the ratio of 4:3 can also be $4(2):3(2)=8:6$ $4(3):3(3)=12:9$ And so on. Marbles are to be removed from a jar that contains 12 red marbles and 12 black marbles. What is the least number of marbles that could be removed so that the ratio of red marbles to black marbles left in the jar will be 4 to 3? *** Right now, there are an equal amount of marbles, so the ratio is 12:12 (or 1:1) We know that we have an end ratio of 4:3 that we want to achieve and that each side of the ratio has to be multiplied (or divided) by the same amount to keep the ratio consistent. We want to remove as few marbles as possible, so let us imagine that 4:3 is a reduced ratio. That means we need to see how many total marbles the reduced ratio of 4:3 could possibly be. So both 4 and 3 have to be multiplied by the same amount to maintain their ratio and yet achieve a higher number of total marbles than just their 7 (4+3=7). We can see that 12 is divisible by 4, so the red marbles could conceivably remain unchanged in order to get a new ratio of 4:3. $12/4=3$ Because 4 can go evenly into 12, this will give us the fewest amount of marbles taken away. Because the 4 is multiplied 3 times to get 12, we know that both 4 and 3 must be multiplied by 3 to keep a new ratio of 4:3 consistent. To find the new number of black marbles, we take 3*3=9. The new amount of black marbles has to be 9. And because our red marbles remain the same (12), we must take only 3 marbles away from the total number of marbles (Why? Because 12â€Å' blackâ€Å' marbles−3 â€Å'blackâ€Å' marbles=9â€Å' blackâ€Å' marbles) So our final answer is 3, we must take 3 black marbles away to get a new ratio of 4:3 of red marbles to black marbles. Finding the Whole If you are given a ratio comparing two parts (piece:anotherâ€Å'piece), and you are told to find the whole amount, simply add all the pieces together. It may help you to think of this like an algebra problem wherein each side of the ratio is a certain multiple of x. Because each side of the ratio must always be divided or multiplied by the same amount to keep the ratio consistent, we can think of each side as having the same variable attached to it. For example, a ratio of 4:5 can be: $4(1):5(1)=4:5$ $4(2):5(2)=8:10$ And so on, just as we did above. But this means we could also represent 4:5 as: $4x:5x$ Why? Because each side must change at the same rate. And in this case, our rate is $x$. So if you were asked to find the total amount, you would add the pieces together. $4x+5x=9x$. The total amount is 9x. In this case, we don’t have any more information, but we know that the total must be divisible by 9. So let’s take a look at another problem. Teyvon has a basket of eggs that he is going to sell. There are two different kinds of eggs in the basket- white and brown. The brown eggs are in a ratio of 2:3 to the white eggs. What is NOT a possible number of eggs that Teyvon can have in the basket? A) 5 B 10 C) 12 D) 30 E) 60 *** In order to find out how many eggs he has total, we must add the two pieces together. So 2+3=5 This means that the total number of eggs in the basket has to either be 5 or any multiple of 5. Why? Because 2:3 is the most reduced form of the ratio of eggs in the basket. This means he could have: $2(2):3(2)=4:6$ eggs in the basket (10 eggs total) $2(3):3(3)=6:9$ eggs in the basket (15 eggs total) And so forth. We don’t know exactly how many eggs he has, but we know that it must be a multiple of 5. This means our answer is C, 12. There is no possible way that he can have 12 eggs in the basket. Now that we are armed with knowledge of fractions and ratios, we must follow the right steps to solve our problems. How to Solve Fraction, Ratio, and Rational Number Questions Now that we have discussed how fractions and ratios work indivisually, let's look at how you'll see them on the test. When you are presented with a fraction or ratio problem, take note of these steps to find your solution: #1: Identify whether the problem involves fractions or ratios A fraction will involve the comparison of a $\piece/\whole$. A ratio will almost always involve the comparison of a piece:piece (or, very rarely, a piece:whole). You can tell when the problem is ratio specific as the question text will do one of three things: Use the : symbol, Use the phrase "___ to ___† Explicitly use the word "ratio† in the text. If the questions wants you to give an answer as a ratio comparing two pieces, make sure you don’t confuse it with a fraction comparing a piece to the whole! #2: If a ratio question asks you to change or identify values, first find the sum of your pieces In order to determine your total amount (or the non-reduced amount of your individual pieces), you must add all the parts of your ratio together. This sum will either be your complete whole or will be a factor of your whole, if your ratio has been reduced. A total of 120,000 votes were cast for 2 opposing candidates, Garcia and Pà ©rez. If Garcia won by a ratio of 5 to 3, what was the number of votes cast for Pà ©rez? (A) 15,000 (B) 30,000 (C) 45,000 D) 75,000 (E) 80,000 *** As you can see, our ratio of 5 to 3 has been greatly reduced (neither of those numbers is in the tens of thousands). We know that there are a total of 120,000 votes, so we need to determine the number of votes for each candidate. Let’s first add our ratio pieces together. 5:3 = 5+3=8 Because 8 is much (much) smaller than 120,000, we know that 8 is not our whole. But 8 is the factor of our whole. ${120,000}/8=15,000$ So if we think of 15,000 as one component (a replacement for our variable, $x$), and Garcia and Pà ©rez have a ratio of 5 components to 3 components, then we can find the total number of votes per candidate. G:P=5:3 = $5x:3x$ 5*15,000=75,000 3*15,000=45,000 So Garcia earned 75,000 votes and Pà ©rez earned 45,000 votes. (You can even confirm that this must be the correct number of votes each by making sure they add up to 120,000. 75,000+45,000=120,000. Success!) So our final answer is C, Pà ©rez earned 45,000 votes. #3: When in doubt, try to use decimals Decimals can make it much easier to work out problems (as opposed to using fractions). So do not be afraid to convert your fractions into decimals to make life easier. #4: Remember your special fractions Always remember that a number over 1 is the same thing as the original number, and that a number over itself = 1. If $h$ and $k$ are positive numbers and $h+k=7$ then ${7-k}/h=$ (A) 1 (B) 0 (C) -1 (D) $h$ (E) $k-1$ *** Here we have two equations: $h+k=7$ and ${7-k}/h$ So let us manipulate the first. $h+k=7$ can be re-written as: $h=7−k$ (Why? We simply subtracted $k$ from either side) So now we can replace the $(7−k)$ from the second equation with $h$, as the two terms are equal. This leaves us with: $h/h$ And we know that any number over itself = 1. So our final answer is A, 1. Now, let's put your knowledge to the test! Test Your Knowledge #1: Flour, water, and salt are mixed by weight in the ratio of 5:4:1, respectively, to produce a certain type of dough. In order to make 5 pounds of this dough, what weight of salt, in pounds, is required? (A) $1/4$ (B) $1/2$ (C) $3/4$ (D) 1 (E) 2 #2: #3: Which of the following answer choices presents the fractions $5/4$, $4/3$, $19/17$, $13/12$, and $7/6$ in order from least to greatest? (A) $19/17$, $7/6$, $13/12$, $4/3$, $7/6$, $5/4$ (B) $4/3$, $5/4$, $7/6$, $19/17$, $13/12$ (C) $13/12$, $7/6$, $19/17$, $5/4$, $4/3$ (D) $19/17$, $13/12$, $5/4$, $7/6$, $4/3$ (E) $13/12$, $19/17$, $7/6$, $5/4$, $4/3$ Answers: B, D, E Answer Explanations: #1: This question is a perfect example of when to find the whole of the pieces of the ratio. Flour, water, and salt are in a ratio of 5:4:1, which means that the whole is: $5x+4x+1x=10x$ So $10x$ is our whole. We want 5 pounds of the recipe, so we must convert $10x$ to 5. $10x=5$ $x=1/2$ Our variable is $1/2$ . Now, we are looking for the amount of salt to use when we started out with $1x$. So let us replace our $x$ with the value we found for it. $1x$ $1(1/2)$ $1/2$ This means we need $1/2$ a pound of salt to make 5 pounds of the mixture. Our final answer is B, $1/2#. #2: For this question, we must find a non-zero integer for t in which $x+{1/x}=t$, where $x$ is also an integer. We know, based on our special fractions, that the only possible way to get a whole number in fraction form is to have our demoninator equal 1 or -1. This means that x cannot possibly be anything other than 1 or negative 1. (Why? If x were anything else but 1, we would end up with a mixed fraction. For example, if x=2, then we would have: $2+{1/2}$. If $x=3$, we would have: $3+{1/3}. And so on. The only way to get an integer value for $t$ is when $x=1$.) So let us try replacing our $x$ value with 1. $x+{1/x}=t$ $1+{1/1}=2$ $t=2$ Well, $t$ could possibly equal 2, but this is not one of our answer choices. So now let us replace $x$ with -1 instead. $x+{1/x}=t$ $-1+{1/-1}=-2$ t=−2 Success! We have found a value for $t$ that matches one of our answer choices. Our final answer is D, $t=−2$ #3: For a problem like this (one that has you order fractions by size), it is usually a good idea to break out the decimals. But we will go through how to solve it using both methods of fractions and decimals. Solving with decimals: To solve with decimals, simply divide each numerator by its denominator to get the decimal. Then, order them in ascending order (as we are told). $5/4=1.25$ $4/3=1.333$ $19/17=1.12$ $13/12=1.08$ $7/6=1.16$ We can see here that the order from least to greatest is: 1.08, 1.12, 1.16, 1.25, 1.33 Which, converted back to their fraction form is: $13/12$, $19/17$, $7/6$, $5/4$, $4/3$ So our final answer is E. Alternatively, we can solve using fractions. Solve using fractions: Let us find a common denominator between all the numerators. A quick way to do this is by multiplying the two largest numerators together. (It may not be the least common denominator, but it'll do for our purposes.) $17*12=204$ Now let's make sure that the other denominators can go evenly into 204 as well. $204/6=34$ $204/4=51$ $204/3=68$ Perfect! Now let us convert all of our fractions. $5/4={5(51)}/{4(51)}=255/204$ $4/3={4(68)}/{3(68)}=272/204$ $19/17={19(12)}/{17(12)}=228/204$ $13/12={13(17)}/{12(17)}=221/204$ $7/6={7(34)}/{6(34)}$ Now that they all share a common denominator, we can compare and order their numerators. So, in ascending order, they would be: $221/204$, $228/204$, $238/204$, $255/204$, $272/204$ Which, when converted back to their original form, is: $13/12$, $19/17$, $7/6$, $5/4$, $4/3$ So again, our final answer is E. I think a nap is in order- don't you? Take-Aways Fractions and ratios may look tricky, but they are merely ways to represent the relationships between pieces of a whole and the whole itself. Once you know what they mean and how they can be manipulated, you’ll find that you can tackle most any fraction or ratio problem the SAT can throw at you. But always remember- though ratios and fractions are related, do not get them mixed up on the SAT! The vast majority of the time, the ratios they give you will compare parts to parts and the fractions will compare parts to the whole. It can be easy to make a mistake during the test, so don’t let yourself lose a point due to careless error. What’s Next? You've conquered fractions and you've decimated ratios and now you're eager for more, right? Well look no further! We have guides aplenty for the many math topics covered on the SAT, including probability, integers, and solid geometry. Feel like you're running out of time on the SAT? Check out our article on how to finish your math sections before time's up. Don't know what score to aim for? Make sure you have a good grasp of what kind of score would best suit your goals and current skill level, and how to improve it from there. Angling to get an 800 on SAT Math? Look to our guide on how to get a perfect score, written by a perfect SAT scorer. Want to improve your SAT score by 160 points? 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