Percent, ratio, and proportion questions on Digital SAT Math test whether you can move fluidly between percentages, decimals, and fractions, and whether you track carefully what quantity a percentage is actually taken of — the original amount, a remaining amount, or a completely different reference quantity. Multi-step problems that chain several percent changes together (a discount followed by another discount, or a series of "percent of a percent" relationships) are where most points are lost, simply from losing track of which base each percentage applies to.
The 22 problems below cover this full range, including compounding percent changes, ratio-and-proportion word problems, multi-step real-world scenarios (webinar attendance, animal composition, map scale), and problems that chain several percentages together through intermediate variables. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. When a problem chains multiple percentages together, it almost always pays to write out each relationship as its own equation before combining them — trying to do it all in your head is where errors creep in. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
If x > 0 and p% of x is 13, which expression represents x in terms of p?
Translate "p% of x is 13" into an equation: (p/100)x = 13.
Solve for x by multiplying both sides by 100/p: x = 13 × (100/p) = (100)(13)/p. The correct answer is D.
What is the 90% of 80?
Convert 90% to a decimal and multiply: 0.90 × 80.
0.90 × 80 = 72. The correct answer is B.
A scientist analyzed a soil sample with a mass of 900 grams and determined that it contained 189 grams of water. What is the percentage of water, by mass, in this soil sample?
Divide the part by the whole: 189/900.
189/900 = 0.21 = 21%. The correct answer is D.
The ratio of green tiles to blue tiles in a piece of artwork is 5 to 2. If there are 16 blue tiles in the piece of artwork, how many green tiles are there?
Set up a proportion: green/blue = 5/2, so green/16 = 5/2.
Solve: green = 16 × (5/2) = 40.
If g/h = 1/100 and 10g/(kh) = 1/100, what is the value of k?
From the first equation, g = h/100. Substitute into the second equation: 10(h/100)/(kh) = 1/100.
The h cancels: 10/(100k) = 1/100. Cross-multiply: 10 × 100 = 100k, so 1,000 = 100k, giving k = 10. The correct answer is B.
The positive quantity r is increased by 20%, and then the resulting quantity is decreased by 25%. Which expression represents the result of these two changes?
A 20% increase multiplies r by 1.20. Applying a 25% decrease to that result multiplies by 0.75: 1.20r × 0.75.
1.20 × 0.75 = 0.90, so the final result is 0.90r. The correct answer is A.
Julia purchased 600 feet of fencing. She used 70% of this fencing to surround a vegetable garden. How many feet of fencing did Julia use to surround the vegetable garden?
Multiply the total fencing by the percentage used: 0.70 × 600.
0.70 × 600 = 420. The correct answer is C.
The ratio 36 to m is equivalent to the ratio 2 to 12. What is the value of m?
Set up the proportion: 36/m = 2/12.
Cross-multiply: 36 × 12 = 2m, so 432 = 2m, giving m = 216.
A clothing store buys shirts at a wholesale price of 5.00 dollars each and resells them at a price that is 270% of the wholesale price. At the end of the season, any remaining shirts are marked at a discounted price that is 70% of the retail price. What is the discounted price of each remaining shirt, in dollars?
Find the retail price first: 270% of $5.00 = 2.70 × 5.00 = $13.50.
Then find 70% of that retail price: 0.70 × 13.50 = 9.45.
A real estate company offers a series of three webinars. 1,250 people attended the first webinar. 68% of the people who attended the first webinar attended the second webinar, and 32% of the people who attended the first and second webinars attended the third webinar. How many people attended all three webinars?
Find attendance at webinar 2: 68% of 1,250 = 0.68 × 1,250 = 850 people.
Then find attendance at webinar 3 (32% of those who attended both 1 and 2): 0.32 × 850 = 272.
If 2a/b = 5.8 and a/(bn) = 23.2, what is the value of 1/n?
From the first equation, a/b = 2.9 (dividing both sides by 2). Rewrite the second equation as (a/b)/n = 23.2, so 2.9/n = 23.2.
Solve for n: n = 2.9/23.2 = 0.125. Then 1/n = 1/0.125 = 8.
The speeds of particles A, B, and C are a meters per second, b meters per second, and c meters per second, respectively. If the speed of particle A is 7,900% of the speed of particle C and the speed of particle C is 0.008% of the speed of particle B, which expression represents the value of a + b in terms of c?
Translate each relationship: a = 7,900% of c = 79c. And c = 0.008% of b = 0.00008b, so b = c/0.00008 = 12,500c.
Adding: a + b = 79c + 12,500c = 12,579c. Note: this works out to 12,579c based on a direct reading of the percentages given, which is 1,000 times larger than answer choice A (12.579c) — if you're using this problem, it's worth double-checking the original percentages (7,900% and 0.008%) against your source, since the scale of the closest answer choice suggests a possible decimal placement difference.
For a certain rectangular region, the ratio of its length to its width is 10 : 4. If the width of the rectangular region increases by 7 units, how must the length change to maintain this ratio?
Let the original width be 4k and length be 10k for some k. After the increase, the new width is 4k + 7. To maintain the 10:4 (or 5:2) ratio, the new length must be (5/2)(4k + 7) = 10k + 17.5.
The change in length is (10k + 17.5) − 10k = 17.5. The correct answer is C.
The composition of an animal is defined as the muscles, bones, and fat of the animal. A scientist studied the composition of one young swamp buffalo and determined the buffalo had 133 kilograms of muscle, which made up approximately 65.9% of its composition. Of the remaining composition of this buffalo, approximately 46.2% was bone, and the remainder was fat. Based on these approximations, to the nearest tenth, how many kilograms of this buffalo's composition was bone?
Find the total composition: 133 ÷ 0.659 ≈ 201.82 kg. The remaining (non-muscle) portion is 201.82 − 133 ≈ 68.82 kg.
Bone is 46.2% of that remaining portion: 0.462 × 68.82 ≈ 31.8 kg.
Raheem bought 9 shirts that were each the same price. He used a coupon for $54 off the entire purchase. The cost for the entire purchase after using the coupon was $108. What was the original price, in dollars, for 1 shirt?
Find the original total before the coupon: $108 + $54 = $162.
Divide by the number of shirts: $162 ÷ 9 = 18.
The combined original price for a towel and chair is $36. After a 45% discount to the chair and a 25% discount to the towel are applied, the combined sale price for the two items is $25. Which system of equations gives the original price t in dollars of the towel and c in dollars of the chair?
The combined original price gives t + c = 36. After discounts, each item is sold for the fraction of its price that remains: the towel (25% off) sells for 75% of t, and the chair (45% off) sells for 55% of c.
That gives 0.75t + 0.55c = 25. The correct answer is D.
After 20% of the original number of objects in a group were removed from the group, 180 objects remained in the group. How many objects were removed?
If 20% were removed, 80% remained: 0.80 × (original) = 180, so the original count was 180/0.80 = 225.
The number removed is 20% of 225: 0.20 × 225 = 45. The correct answer is B.
When a collection of books is sorted, 52% are nonfiction and the remaining books are fiction. Of those that are fiction, 75% are westerns and the remaining fiction books are romance. If there are 933 romance books in this collection, what is the total number of books in the collection?
Romance books make up 25% of the fiction books (since 100% − 75% = 25%): 0.25 × (fiction) = 933, so fiction = 933/0.25 = 3,732.
Fiction books make up 48% of the total (since 100% − 52% = 48%): 0.48 × (total) = 3,732, so total = 3,732/0.48 = 7775.
The positive number a is 2,800% of the number c, and c is 20% of the number b. What is the value of a − b in terms of c?
Translate each relationship: a = 2,800% of c = 28c. And c = 20% of b = 0.20b, so b = c/0.20 = 5c.
Then a − b = 28c − 5c = 23c. The correct answer is A.
For each of two rectangles, the ratio of the rectangle's length to its width is 10 to 5. The width of rectangle X is 16.5 times the width of rectangle Y. How does the length of rectangle X compare to the length of rectangle Y?
Since both rectangles share the same length-to-width ratio (10:5, or 2:1), length and width scale together proportionally — whatever factor relates the widths also relates the lengths.
Since the width of X is 16.5 times the width of Y, the length of X must also be 16.5 times the length of Y. The correct answer is C.
A square map has a side length of 35 inches, and 1 inch on the map represents an actual distance of 17 miles. A smaller version of the same map is printed as a square with the side length 55% shorter than the side length of the previous map. On the smaller map, which of the following is closest to the actual distance, in miles, represented by 1 inch?
The original map's full 35-inch side represents 35 × 17 = 595 actual miles. The smaller map is 55% shorter, so its side length is 35 × (1 − 0.55) = 35 × 0.45 = 15.75 inches — but it still represents the same 595 miles across that side, since it's just a smaller print of the same region.
So the new scale is 595 miles ÷ 15.75 inches ≈ 37.78 miles per inch. The correct answer is D.
For the positive quantities h, j, and k: 94% of h is equivalent to 47% of j, and j is equivalent to 30% of k. What percentage of k is h? (Disregard the % sign when entering your answer. For example, if your answer is 39%, enter 39)
Translate the first relationship: 0.94h = 0.47j, so j = 0.94h/0.47 = 2h.
From the second relationship, j = 0.30k. Substitute: 2h = 0.30k, so h = 0.15k — meaning h is 15% of k. Enter 15.
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