Digital SAT Math · Problem-Solving & Data Analysis
Ratios, rates & proportions
Digital SAT Math · Problem-Solving and Data Analysis
A ratio item hands you a comparison and asks you to carry it to a different size. The arithmetic is one multiplication or one division; the entire difficulty is deciding which, and to what. Write the units into every cell of the setup and the decision makes itself.
On the test
| Domain | Problem-Solving and Data Analysis (score report) |
| What it looks like | A comparison — parts of a mixture, a price per gram, a scale on a drawing, a table of paired values — plus one new amount to carry it to |
| Often asked | “At this rate, how many…?”, “What is the total number of…?”, “How many must be added…?” |
| Format | Multiple choice and student-produced response |
| Calculator | Allowed throughout; the numbers are usually calculator-easy, which is why the setup carries all the risk |
| Figures | About one item in five arrives with a small value table; graphs and diagrams belong to other skills |
Recognition cues: for every, per, at this rate, in the ratio, proportional to, represents, scale.
Pattern recognition
Every item on this skill is one comparison stretched or shrunk to a new size. What changes is which quantity is missing.
- Rate forward — a rate is given, the new amount is given, the total is missing. “96 jars in 8 minutes; how many in 15?”
- Rate backward — the total is given and the input is missing. “How many gallons for 390 miles?” Same rate, opposite operation.
- Parts of a whole — a part-to-part ratio plus one absolute number (a total, or a difference between the parts).
- Proportional table or scale — the comparison is delivered as a table of paired values or a drawing scale, and you must extract the constant yourself.
Method
- Name the two quantities and their units. Jars and minutes, grams and dollars, cups of concentrate and cups of water.
- Write the given comparison as a labeled fraction. From a table, take any one row — in a proportional table every row gives the same fraction, and checking a second row is your free accuracy check.
- Reduce it to a unit rate if that helps you think. jars per minute; dollars per gram. A unit rate is just the comparison with a 1 in the denominator, and it makes the next step a single multiplication.
- Set the second fraction with the labels in the same positions, put where the unknown belongs, and cross-multiply.
- Unit-check the answer. Multiplying dollars-per-gram by grams leaves dollars; if your leftover unit is not the one the question asked for, the setup was upside down.
- Answer the asked quantity. Part or whole? Added amount or final amount? This is where correct arithmetic most often gets thrown away.
| Words in the stem | What they mean |
|---|---|
| at this rate, at the same rate | the comparison stays constant — build a proportion |
| the ratio of A to B is 3 to 5 | part-to-part: the whole is 3 + 5 = 8 shares |
| 3/5 of the saplings | part-to-whole: the whole is already the denominator |
| directly proportional to | ; find from the given pair before anything else |
| per 100 grams, per dozen | the rate is stated for a block of units, not for one — scale it |
| represents, scale of | a ratio between a drawing and reality, with different units on each side |
| how many more must be added | the answer is a change, not the new total |
Worked example 1 — the labeled proportion
Stem. A workshop uses 6 metres of ribbon to trim 4 aprons. At this rate, how many metres of ribbon are needed to trim 26 aprons?
Step 1 — name and label. Metres of ribbon on top, aprons on the bottom.
Step 2 — set the proportion.
Step 3 — cross-multiply.
Check. The unit rate is metres per apron, and . Leftover unit: metres, which is what was asked. Answer: 39 metres.
Trap watch. Flipping the first fraction to gives metres — less ribbon for six times as many aprons. Adding instead of scaling gives .
Worked example 2 — part-to-part with a total
Stem. A collection contains only domestic and foreign stamps, in the ratio 4 to 7. There are 176 stamps in all. How many are foreign?
Step 1 — count the shares. shares make up the whole.
Step 2 — size one share.
Step 3 — take the shares you need. Foreign holds 7 shares: .
Check. Domestic is , and . The ratio reduces to . Answer: 112 foreign stamps.
Trap watch. treats a part-to-part ratio as a fraction of the whole — and a non-integer stamp count is the tell. Answering 64 solves the problem and then reports the other part.
Worked example 3 — a rate stated per block of units
Stem. A spreader applies 1.8 kilograms of fertiliser per 500 square feet of lawn. A lawn measures 3,250 square feet. How much fertiliser does it need?
Step 1 — notice the block. The rate is per 500 square feet, not per square foot. That gap is the whole item.
Step 2 — find the scaling factor.
Step 3 — scale the rate.
Check. The per-square-foot rate is , and . Same answer by a longer road. Answer: 11.7 kilograms.
Trap watch. Answering 1.8 is the skipped-scaling error — reading the label and reporting it. Dividing the wrong way, , gives 277.8, a number with no meaning in this problem at all.
Practice
Answer before opening the explanation. Two items are student-produced response (type the number, no choices), and two arrive with a value table, the way roughly one ratio item in five does on the real test. Every wrong choice below is a specific error with a name — log the name when you miss one.
Question 1 Warm-up
A bottling line at an apiary fills 96 jars of honey in 8 minutes. At this rate, how many jars does the line fill in 15 minutes?
Show the answer Choice D
Why it is right
Write the rate with its labels: 96 jars per 8 minutes, which is 96 / 8 = 12 jars per minute. Multiply by the new time: 12 jars per minute times 15 minutes = 180 jars, and the minute labels cancel to leave jars, which is what the question asks for. The proportion form gives the same thing: 96 jars / 8 minutes = x jars / 15 minutes, so 8x = 96 times 15 = 1,440 and x = 180. Sanity check the size: 15 minutes is almost twice 8 minutes, so the answer must be a little under twice 96.
Why each other choice fails
- Choice A
- The proportion was built with the labels swapped: 96/8 = 15/x gives x = 8 x 15 / 96 = 1.25. That is a minutes-per-jar calculation dressed up as an answer, and it says the line slows from 12 jars a minute to about one jar per quarter hour.
- Choice B
- 12 is the unit rate in jars per minute, the intermediate value, not the total for 15 minutes. It still has to be multiplied by the 15 minutes the question asks about.
- Choice C
- Additive instead of multiplicative: 15 minutes is 7 more than 8 minutes, so 7 more jars were added to 96. Rates scale by multiplying, not by adding the same number to both quantities — 7 extra minutes buy 84 extra jars, not 7.
Question 2 Standard
A nursery grows only oak and maple saplings. The ratio of oak saplings to maple saplings is 3 to 5, and the nursery has 240 saplings in all. How many of the saplings are oaks?
Show the answer Choice B
Why it is right
The ratio 3 to 5 is part-to-part, so the whole is 3 + 5 = 8 equal shares. Divide the total by the number of shares: 240 / 8 = 30 saplings per share. Oaks take 3 shares, so oaks = 3 x 30 = 90. Equivalently, oaks are 3/8 of the whole: (3/8)(240) = 90. Check both parts: maples are 5 x 30 = 150, and 90 + 150 = 240, matching the stated total, with the oak-to-maple ratio 90:150 reducing to 3:5.
Why each other choice fails
- Choice A
- 30 is the size of one share, the number you get after dividing 240 by 8. It is a real step in the work, but oaks occupy three of those shares, not one.
- Choice C
- Treats the part-to-part ratio 3:5 as if it were the part-to-whole fraction 3/5, giving (3/5)(240) = 144. The 5 counts maples, not the total; only 3 + 5 = 8 counts the total.
- Choice D
- 150 is the number of maples, (5/8)(240). The arithmetic is right and the quantity is wrong — the question asked for oaks, the smaller of the two parts.
Question 3 Standard
A cereal label states that the cereal contains 12 grams of protein per 100 grams of cereal. One serving of the cereal has a mass of 40 grams. How many grams of protein are in one serving?
Show the answer Choice A
Why it is right
The label gives a rate, 12 grams of protein per 100 grams of cereal, and the serving is smaller than the 100 grams the rate is stated for, so the protein must be scaled down. Set up the labeled proportion 12 g protein / 100 g cereal = x g protein / 40 g cereal, which gives 100x = 480 and x = 4.8 grams. The scaling view is the same: a 40-gram serving is 40/100 = 0.4 of the reference amount, and 0.4 x 12 = 4.8. Check that the answer is smaller than 12, as any part of a 100-gram figure must be.
Why each other choice fails
- Choice B
- Copies the number off the label without scaling it. 12 grams is the protein in 100 grams of cereal, and a serving is only 40 grams — the per-100-gram figure is never the per-serving figure unless the serving happens to be 100 grams.
- Choice C
- Inverts the proportion: 12 x 100 / 40 = 30 puts the serving size in the denominator instead of the numerator. That would mean a 40-gram serving holds more protein than 100 grams of the same cereal.
- Choice D
- A decimal slip that reads '12 grams per 100 grams' as 1.2 grams of protein per gram of cereal, then multiplies 40 x 1.2 = 48. The true rate is 0.12 grams per gram, and 48 grams of protein cannot fit inside a 40-gram serving.
Question 4 Standard
A silversmith prices each finished piece in proportion to the mass of sterling silver it contains. The table shows the mass and the price of three pieces. At this rate, what is the price of a piece containing 96 grams of sterling silver?
| Mass (grams) | Price (dollars) |
|---|---|
| 16 | 20 |
| 28 | 35 |
| 52 | 65 |
Show the answer Choice D
Why it is right
Because price is proportional to mass, every row of the table has the same price-to-mass ratio, and that ratio is the constant of proportionality. From the first row, 20 / 16 = 1.25 dollars per gram; the other rows agree, since 35 / 28 = 1.25 and 65 / 52 = 1.25. Multiply the rate by the new mass: 1.25 dollars per gram x 96 grams = $120.00, and the gram labels cancel to leave dollars. Check the size: 96 grams is a little under twice the 52-gram piece, and $120 is a little under twice $65.
Why each other choice fails
- Choice A
- Prices only the extra silver beyond the last row of the table: 96 - 52 = 44 grams, and 1.25 x 44 = 55. The $65 already earned by the first 52 grams was dropped, so this is the increase in price, not the price.
- Choice B
- Divides by the rate instead of multiplying by it: 96 / 1.25 = 76.8. Dollars per gram must multiply a mass to produce dollars; dividing by it would answer 'how many grams cost $96'.
- Choice C
- Additive instead of multiplicative: 96 grams is 44 more than the 52-gram row, so $44 was added to $65. Proportional quantities scale by a common factor, not by matching increments — 44 extra grams cost $55, not $44.
Question 5 Standard
On a highway trip, a car traveled 234 miles using 9 gallons of fuel. At this rate, how many gallons of fuel are needed to travel 390 miles?
Show the answer Choice A
Why it is right
First get the rate from the trip that is fully described: 234 miles / 9 gallons = 26 miles per gallon. The question asks for gallons, so divide the new distance by that rate: 390 miles / 26 miles per gallon = 15 gallons, and the mile labels cancel to leave gallons. The labeled proportion does the same work in one line: 234 miles / 9 gallons = 390 miles / x gallons, so 234x = 3,510 and x = 15. Check the direction: 390 miles is well under twice 234 miles, and 15 gallons is under twice 9 gallons.
Why each other choice fails
- Choice B
- 26 is the fuel economy in miles per gallon, not a number of gallons. It is the rate you build on the way to the answer; the answer still needs 390 divided by it.
- Choice C
- Divides the new distance by the old fuel amount: 390 / 9 = 43.3. That mixes a mileage from one trip with a fuel amount from another and produces miles per gallon for a trip nobody took.
- Choice D
- Additive instead of multiplicative: 390 miles is 156 more than 234, so 156 was added to the 9 gallons. Fuel use scales with distance by a factor — those extra 156 miles need 6 gallons, not 156.
Question 6 Harder
A caterer makes punch by mixing concentrate and water in the same ratio for every batch. The table shows the amounts used in three batch sizes. If the caterer makes a batch using 36 cups of water, how many cups of punch does the batch contain?
| Concentrate (cups) | Water (cups) |
|---|---|
| 2 | 6 |
| 5 | 15 |
| 9 | 27 |
Show the answer Choice B
Why it is right
Every row of the table has the same concentrate-to-water ratio: 2:6, 5:15 and 9:27 all reduce to 1:3, so there are 3 cups of water for every 1 cup of concentrate. With 36 cups of water, the concentrate is 36 / 3 = 12 cups. The punch is the mixture, so its volume is the whole, not either part: 12 + 36 = 48 cups. Check against the ratio: 12:36 reduces to 1:3, matching every row of the table.
Why each other choice fails
- Choice A
- 12 is the concentrate alone. It is the right first step, but the question asks how much punch the batch contains, and punch is concentrate plus water — the classic part-instead-of-whole miss.
- Choice C
- Uses the ratio upside down, multiplying the water by 3 instead of dividing: 36 x 3 = 108. The table says water is the larger part, so the concentrate must come out smaller than 36, not triple it.
- Choice D
- Same inversion as C, then totalled: 108 + 36 = 144. The sum is the right final move applied to a wrong part, which is why the answer is exactly three times the correct 48.
Question 7 Harder
The mass of a spool of steel cable is directly proportional to the length of cable on the spool. A spool holding 6 meters of cable has a mass of 15 kilograms. What is the mass, in kilograms, of a spool holding 22 meters of the same cable?
Show the answer Choice C
Why it is right
Directly proportional means mass = k x length for a single constant k, so read k off the pair you are given: k = 15 kg / 6 m = 2.5 kilograms per meter. Then mass = 2.5 x 22 = 55 kilograms, and the meter labels cancel to leave kilograms. The proportion form does the same work: 15 / 6 = x / 22, so 6x = 330 and x = 55. Check the size before moving on: 22 meters is between three and four times 6 meters, so the mass must land between 45 and 60 kilograms.
Why each other choice fails
- Choice A
- Uses the constant upside down: 6 / 15 = 0.4 is meters per kilogram, and 22 x 0.4 = 8.8. That makes a 22-meter spool lighter than the 6-meter one, which no proportional relationship with a positive constant allows.
- Choice B
- Additive instead of multiplicative: 22 meters is 16 more than 6 meters, so 16 kilograms were added to 15. Proportional quantities scale by a common factor — those 16 extra meters weigh 40 kilograms, not 16.
- Choice D
- 330 is the cross product 15 x 22, left unfinished. Solving 15/6 = x/22 requires dividing that product by 6; stopping early leaves an answer six times too large.
Question 8 Harder Student-produced response
On an architectural drawing, a length of 2 inches represents an actual length of 9 feet. A hallway is drawn 15 inches long. What is the actual length of the hallway, in feet?
Show the answer 67.5
Why it is right
A drawing scale is a ratio with two different units, so label every cell: 2 inches on the drawing per 9 feet in the building. The scale factor is 9 / 2 = 4.5 feet per inch of drawing, so 15 inches represents 15 x 4.5 = 67.5 feet. The proportion form is the same: 2 in / 9 ft = 15 in / x ft, giving 2x = 135 and x = 67.5. Check the direction: 15 inches is 7.5 times 2 inches, and 67.5 feet is 7.5 times 9 feet.
Answers students type instead
- 22
- Additive instead of multiplicative: 9 feet is 7 more than 2 inches, so 7 was added to 15. Scales multiply, and the numbers being added are not even in the same units.
- 30
- Multiplies the drawing length by 2 because 2 appears in the scale. The 2 is the drawing side of the ratio, not the scale factor; the factor is 9/2 = 4.5.
- 3.3
- Builds the proportion with the labels swapped, 2/9 = x/15, which gives 30/9 = 3.33. That answers 'how many inches of drawing represent 15 feet', and no hallway drawn at 15 inches is barely 3 feet long.
Question 9 Harder
In a museum's collection, the ratio of prints to paintings is 3 to 2, and the ratio of paintings to sculptures is 5 to 4. If the collection contains 120 sculptures, how many prints does it contain?
Show the answer Choice D
Why it is right
Two ratios can only be chained through the quantity they share, which here is paintings. Work from the sculptures inward: paintings / sculptures = 5/4, so paintings = (5/4)(120) = 150. Then prints / paintings = 3/2, so prints = (3/2)(150) = 225. As a single chain, prints = 120 x (5/4) x (3/2) = 225. Check the whole set: 225 prints, 150 paintings and 120 sculptures give 225:150 = 3:2 and 150:120 = 5:4, both as stated.
Why each other choice fails
- Choice A
- Multiplies 120 by 3/4, pairing the 3 from the first ratio with the 4 from the second as though they described the same comparison. The two ratios share only paintings, so the 3 and the 4 never meet directly.
- Choice B
- 150 is the number of paintings, the middle quantity you pass through on the way to the answer. It is correct arithmetic answering the wrong question.
- Choice C
- Applies the prints-to-paintings ratio straight to the sculptures: (3/2)(120) = 180. That skips the conversion from sculptures to paintings, which is exactly the step the second ratio was given for.
Question 10 Hardest
Working at the same constant rate, 3 sewing machines produce 2,400 tote bags in 5 hours. At that same rate per machine, how many tote bags do 5 machines produce in 6 hours?
Show the answer Choice D
Why it is right
Reduce the given information to a rate per machine per hour, because that is the only quantity both scenarios share. The first scenario supplies 3 x 5 = 15 machine-hours for 2,400 bags, so the rate is 2,400 / 15 = 160 bags per machine-hour. The second scenario supplies 5 x 6 = 30 machine-hours, so the output is 160 x 30 = 4,800 bags. Check with scale factors instead: machines go up by 5/3 and hours by 6/5, and 2,400 x (5/3) x (6/5) = 4,800, the same answer.
Why each other choice fails
- Choice A
- Scales by 3/5 instead of 5/3, inverting the machine ratio, and ignores the change in hours: 2,400 x (3/5) = 1,440. More machines running longer cannot produce fewer bags than the original setup.
- Choice B
- Scales the hours and forgets the machines: 2,400 x (6/5) = 2,880. That is the output of the original 3 machines running 6 hours, not of 5 machines.
- Choice C
- Scales the machines and forgets the hours: 2,400 x (5/3) = 4,000. That is what 5 machines make in the original 5 hours. Both factors have to be applied, since output is proportional to machines and to time.
Question 11 Hardest Student-produced response
A campus snack cart stocks granola bars and fruit cups in the ratio 7 to 5. The cart holds 24 more granola bars than fruit cups. What is the total number of granola bars and fruit cups on the cart?
Show the answer 144
Why it is right
Write both parts in shares: granola bars are 7k and fruit cups are 5k for the same share size k. The stem describes a difference, so translate it: 7k - 5k = 24, meaning 2k = 24 and k = 12 items per share. The total is 7k + 5k = 12k = 12 x 12 = 144. Check: granola bars are 7 x 12 = 84, fruit cups are 5 x 12 = 60, the difference is 84 - 60 = 24 as stated, and 84:60 reduces to 7:5.
Answers students type instead
- 60
- The number of fruit cups, 5k. Same wrong-quantity miss as 84, on the smaller part.
- 84
- The number of granola bars, 7k. It is one of the two parts, and the question asked for the total of both.
- 288
- Treats the 24 as the size of one share and multiplies by the 12 total shares. The 24 is the gap between the parts, which is 2 shares wide, so one share is 12, not 24.
Question 12 Hardest
A muralist has 60 ounces of mixed paint in which the ratio of blue paint to white paint is 2 to 3. How many ounces of white paint must be added to the mixture so that the ratio of blue paint to white paint becomes 1 to 2?
Show the answer Choice A
Why it is right
Start by splitting the current mixture into shares: 2 + 3 = 5 shares, so each share is 60 / 5 = 12 ounces, giving 24 ounces of blue and 36 ounces of white. Only white is added, so the blue stays at 24 ounces. The target ratio 1 to 2 means white must end up twice the blue, that is 2 x 24 = 48 ounces. The amount to add is the change, not the final amount: 48 - 36 = 12 ounces. Check: the new mixture is 24 blue and 48 white, and 24:48 reduces to 1:2.
Why each other choice fails
- Choice B
- Reads the target ratio backwards, as if white had to be half the blue: that would call for 12 ounces of white, a change of 36 - 12 = 24 ounces in the wrong direction. Adding paint can only raise the white total.
- Choice C
- 36 is the white already in the mixture, computed correctly from the 2:3 split but never compared with the target. It is the starting value, not the amount added.
- Choice D
- 48 is the white the mixture must finish with. The question asks how much is added, so the 36 ounces already present still have to be subtracted.
Common mistakes
- Inverted proportion — the second fraction built with the labels in swapped cells, so the answer comes out reciprocal-sized. The unit check catches every instance of this.
- Part-to-part read as part-to-whole — “3 to 5” turned into of the total instead of . The most-named error on this skill, and the most-missed.
- Adding instead of scaling — 22 metres treated as “16 more than 6”, so 16 is added to the mass. Proportional means a common factor, never a common difference.
- Stopping at the unit rate — computing 12 jars per minute or 26 miles per gallon and grid-in-ing that instead of the quantity asked for.
- Skipping the block scaling — a rate given per 100 grams or per 500 square feet copied straight into the answer, as though the block were one unit.
- Answering the part when the whole was asked (or the reverse) — the concentrate rather than the punch, one colour rather than the mixture.
- Reporting the final amount when the question asked for the change — “how many ounces must be added” is a difference, not the new total.
- Chaining two ratios through the wrong term — and only connect through ; pairing the outer numbers directly is not a legal move.
- Dropping units mid-solution — the mistake that makes all of the above invisible. Units in every cell is not decoration; it is the error check.
FAQ
Is a proportional relationship the same as a linear one? Every proportional relationship is linear, but not the reverse. Proportional means — the graph passes through the origin and the ratio is the same for every pair. A table with a start-up fee, like , is linear with a constant difference but no constant ratio, so proportional reasoning on it gives wrong answers.
Do I need to convert units before setting up? Only when the same quantity appears in two units in one comparison. A rate can happily compare unlike units — miles per gallon, dollars per gram, feet per inch of drawing. Mixed units for the same quantity (minutes against hours, inches against feet) must be reconciled first; those chain-conversion items are the neighbouring skill.
What about scale factors and area? If every length of a figure is multiplied by , the area is multiplied by and the volume by . It is proportional reasoning, but the SAT files it under similarity, so it is only mentioned here.
How do I enter a ratio in a grid-in? You never enter a ratio — SPR answers are numbers. If the work ends at “7 to 5”, the stem is asking for a count, a length or a total, so carry the ratio one step further and enter that. Reread the last line before typing: on this skill the wrong-quantity error is the expensive one.
Coming next in this domain: units and conversions.