A ratio preserves a relationship between quantities. It does not tell you their actual sizes until you know a total or one of the amounts. Write the labels beside the numbers so the relationship stays in the right order.
Test Ninjas Research · Updated September 29, 2026
What you will practice
Keep each ratio quantity in the stated order.
Separate a part-to-part ratio from a fraction of the whole.
Choose a scale factor or unit rate and keep its units.
Suggested pace: 10 minutes on examples · 10–15 minutes on questions · 5 minutes on review. Take a second session if a method needs more work.
Keep in mind: For a ratio a:b, the combined whole has a + b equal shares.
Method 1 of 4
Distinguish part-to-part from part-to-whole
If red to blue is 2:3, there are two red shares for every three blue shares. Red is two out of five combined shares, not two out of three. A simple bar divided into equal shares can make this distinction visible.
Label the ratio in its stated order.
Add the ratio numbers when the question gives a combined total.
Find the size of one share, then multiply by the required number of shares.
Worked example
Red and blue tiles are in the ratio 2:3. There are 40 tiles altogether. How many are red?
The whole contains 2 + 3 = 5 shares.
One share contains 40 ÷ 5 = 8 tiles.
Red tiles: 2 × 8 = 16. Blue tiles: 3 × 8 = 24.
Answer: 16 red tiles
Check it: 16 + 24 = 40, and 16:24 simplifies to 2:3.
Now try it without looking back
Red to blue is 1:4. What fraction of all the tiles is red?
Check your reasoning
1/5. There are 1 + 4 = 5 total shares, of which one is red.
Common mistake: Calculating 2/3 of 40 confuses red-to-blue with red-to-total.
Method 2 of 4
Scale from one known quantity
Equivalent ratios multiply or divide both terms by the same factor. If one quantity is known, compare it with its matching ratio term; do not divide by the total number of shares unless the known number is the combined total.
Match the known amount to the correct side of the ratio.
Find the common scale factor.
Apply that factor to the other side.
Worked example
A mix uses 3 cups of concentrate for every 5 cups of water. How much water is needed with 12 cups of concentrate?
The concentrate amount is 12 ÷ 3 = 4 times the recipe.
The water must also be multiplied by 4.
Water needed = 5 × 4 = 20 cups.
Answer: 20 cups of water
Check it: Both quantities are four times their starting amounts; the mix has not become stronger or weaker.
Now try it without looking back
Concentrate to water is 2:5. How much water goes with 6 cups of concentrate?
Check your reasoning
15 cups. The scale factor is 6 ÷ 2 = 3, so multiply 5 cups of water by 3.
Common mistake: Adding 9 cups to both ingredients preserves a difference, not a ratio.
Method 3 of 4
Use a unit rate to compare or scale
A unit rate puts one quantity over a single unit of another: dollars per item, miles per hour, or inches per foot. Track the units through the division. A lower package price is not necessarily a lower price per item.
Divide to find the amount per one unit.
Compare like units or multiply by the requested quantity.
Keep the final units in the answer.
Worked example
A pack of 8 batteries costs $12. A pack of 12 costs $16.80. Which has the lower price per battery?
First pack: $12 ÷ 8 = $1.50 per battery.
Second pack: $16.80 ÷ 12 = $1.40 per battery.
The second pack costs $0.10 less per battery.
Answer: The 12-pack
Check it: Twelve batteries at $1.50 each would cost $18, which is more than $16.80.
Now try it without looking back
Six items cost $9. What would ten cost at the same unit price?
Check your reasoning
$15. First find $9 ÷ 6 = $1.50 per item, then multiply by ten.
Common mistake: A comparison is meaningful only when the denominators describe the same thing. Convert ounces to pounds, or minutes to hours, before comparing unlike units.
Method 4 of 4
Decide whether the relationship is direct or inverse
More identical items cost more at a fixed unit price: that is a direct relationship. More equally productive workers need less time for a fixed job: that is inverse, provided the work can be shared without interference. Think about the expected direction before setting up a proportion.
Identify what stays fixed.
Ask whether increasing one quantity should increase or decrease the other.
Use equivalent ratios for direct scaling, or a constant product for inverse scaling.
Worked example
Four equally productive workers complete a shareable job in 6 hours. How long would 8 workers take at the same individual rate?
The job requires 4 × 6 = 24 worker-hours.
Eight workers supply 8 worker-hours each hour.
Time = 24 ÷ 8 = 3 hours.
Answer: 3 hours
Check it: Doubling the workforce halves the time under the stated assumptions.
Now try it without looking back
Two identical pumps empty a tank in 12 hours. How long would four take at the same rate?
Check your reasoning
6 hours, assuming their work combines without interference. Twice as many pumps take half as long.
Common mistake: Doubling both workers and hours would make the larger group do four times as much work, not the same job.
Practice ratios and proportions
Write a word above each side of your ratio. When you finish, check both the ratio and any given total; satisfying only one is not enough.
Work the first six questions with the explanations closed. Review any missed or guessed answers before continuing with the other six. If you cannot set up a problem, return to its method above and try again from a blank page.
Ratios and proportions: 12 practice questions
Work through one question at a time. Explain your approach, check the worked answer, then move on. Use scratch paper for calculations.
0 of 12 checked · 0 correct.
Progress is saved in this browser. These original study questions do not produce an official AFQT score.
Question 1: Completing a three-part mixture
Question 1 · Completing a three-part mixture · easy
A soil mixture calls for peat, compost, and grit in the volume ratio 2:3:1. A gardener has already combined 12 liters of peat with 18 liters of compost. How many liters of grit should be added to complete this mixture?
Worked answer and explanation
Answer: B. 6 liters
Two ratio parts of peat equal 12 liters, so one part equals 6 liters. The compost confirms the scale because 3 × 6 = 18 liters. Grit accounts for one part, so add 6 liters. Taking one sixth of the compost alone would give 3 liters, but the six total ratio parts describe the entire mixture.
Question 2: Ratio after removing one color
Question 2 · Ratio after removing one color · medium
A box contains only red and white tokens in a ratio of 5 red to 3 white. After 12 red tokens are removed, the box contains equal numbers of the two colors. How many tokens remain in the box?
Worked answer and explanation
Answer: C. 36
The original red excess is two ratio parts, and removing 12 eliminates that excess. One part is therefore 12 ÷ 2 = 6 tokens. The 3 white parts contain 18 tokens; 18 red tokens remain as well. The remaining total is 36. Forty-eight is the original total before removal.
Question 3: Inverse mixture concentration
Question 3 · Inverse mixture concentration · medium
A technician mixes a 10% salt solution and a 30% salt solution to make 12 liters of 15% salt solution. Percentages mean salt amount per unit volume, and the solution volumes add with no loss. How many liters of the 30% solution are needed?
Worked answer and explanation
Answer: A. 3 liters
Using all 10% solution would supply 0.10 × 12 = 1.2 units of salt, but the target needs 0.15 × 12 = 1.8. Each liter replaced by 30% solution adds 0.30 − 0.10 = 0.20 unit of salt. The required amount is (1.8 − 1.2) ÷ 0.20 = 3 liters. Nine liters is the amount of the 10% solution.
Question 4: A ratio changed by new members
Question 4 · A ratio changed by new members · hard
A club initially has boys and girls in the ratio 4:5. Nine girls join, and no one else joins or leaves. The new ratio of boys to girls is 2:3. How many members did the club have before the nine girls joined?
Worked answer and explanation
Answer: C. 81
Let the original counts be 4k and 5k. The new ratio gives 4k/(5k + 9) = 2/3, so 12k = 10k + 18 and k = 9. Initially there were 9k = 81 members. The 90 choice includes the nine newcomers, whereas the question asks for the original membership.
Question 5: Water per planted container
Question 5 · Water per planted container · easy
A gardener divides 12 liters of water equally among 18 planted containers. How many liters of water does each container receive?
Worked answer and explanation
Answer: B. 2/3 liter
Divide the total water by the number of containers: 12/18 = 2/3 liter per container. The reciprocal 18/12 = 1 1/2 describes containers per liter, not liters per container. Subtracting 12 from 18 does not give an amount per container.
Question 6: Ratio shares from a count difference
Question 6 · Ratio shares from a count difference · medium
A tray contains screws and washers in the ratio 9 screws to 4 washers. There are 35 more screws than washers. How many washers are in the tray?
Worked answer and explanation
Answer: A. 28
The difference is 9 − 4 = 5 ratio shares, so one share is 35/5 = 7 items. Washers occupy four shares: 4 × 7 = 28. Sixty-three is the screw count, and 91 is the combined count; neither is the requested washer count.
Question 7: Correcting the amount of one mixture component
Question 7 · Correcting the amount of one mixture component · medium
A container holds 4 liters of concentrate and 14 liters of water. The desired mixture has 3 liters of concentrate for every 7 liters of water. Only concentrate will be added, with no loss of either liquid. How many liters of concentrate should be added?
Worked answer and explanation
Answer: A. 2 liters
Fourteen liters of water is twice the 7-liter ratio amount, so the mixture needs 2 × 3 = 6 liters of concentrate altogether. Four liters are already present, leaving 6 − 4 = 2 liters to add. Six liters is the desired total concentrate, not the additional amount.
Question 8: A recipe limited by one ingredient
Question 8 · A recipe limited by one ingredient · hard
A dry grain blend uses rice, lentils and barley in the mass ratio 5:3:2. Available supplies are 4.5 kilograms of rice, 2.1 kilograms of lentils and 1.8 kilograms of barley. No ingredient may be substituted for another. What is the greatest number of full 0.25-kilogram bags that can be filled with the correctly proportioned blend?
Worked answer and explanation
Answer: B. 28 bags
Rice could support 4.5/(5/10) = 9 kilograms of blend, lentils 2.1/(3/10) = 7 kilograms, and barley 1.8/(2/10) = 9 kilograms. Lentils limit the blend to 7 kilograms, enough for 7/0.25 = 28 full bags. Adding all 8.4 kilograms of stock and dividing by bag size gives 33 full bags, but that stock would not have the required proportions.
Question 9: Perimeter after a linear scale reduction
Question 9 · Perimeter after a linear scale reduction · easy
A square display tile has sides 8 inches long. A scale model uses 1 inch for every 4 inches of the actual tile. What is the perimeter of the square tile in the model?
Worked answer and explanation
Answer: C. 8 inches
Each model side is 8/4 = 2 inches long. A square has four equal sides, so the model perimeter is 4 × 2 = 8 inches. Two inches is one model side, and 32 inches is the original tile’s perimeter before scaling.
Question 10: Combining unequal ratio groups
Question 10 · Combining unequal ratio groups · medium
One box contains 21 wristbands with red to blue in the ratio 2:5. A second box contains 18 wristbands with red to blue in the ratio 5:4. If the boxes are combined, what fraction of all the wristbands are red?
Worked answer and explanation
Answer: B. 16/39
The first box has 21 × 2/7 = 6 red bands, and the second has 18 × 5/9 = 10. Together they contain 16 red bands out of 39 total, giving 16/39. Averaging the two red shares without weighting their different box totals does not give the combined share.
Question 11: Removing equal counts from unequal ratio groups
Question 11 · Removing equal counts from unequal ratio groups · medium
A box initially contains blue and red tokens in the ratio 3:2. After 12 blue tokens and 12 red tokens are removed, the remaining ratio of blue to red is 3:1. How many red tokens were in the box originally?
Worked answer and explanation
Answer: D. 16
Let the original counts be 3 k and 2 k. The final ratio requires 3 k − 12 = 3(2 k − 12), giving 3 k = 24 and k = 8. There were 2 k = 16 red tokens originally. Four is the remaining red count; the question asks the original count. Equal subtractions change a ratio because they remove different fractions of its unequal groups.
Question 12: Comparing opposite transfers between two mixtures
Question 12 · Comparing opposite transfers between two mixtures · hard
Container A starts with 10 liters of clear liquid, and container B starts with 10 liters of dyed liquid. Two liters from A are poured into B and mixed uniformly. Then 2 liters of that mixture are poured back into A. Assume volumes add and none is lost. After both transfers, what is the ratio of the volume of A’s original liquid now in B to the volume of B’s original liquid now in A?
Worked answer and explanation
Answer: C. 1:1
After the first transfer, B contains 2 liters of A’s original liquid and 10 liters of B’s original liquid, for 12 liters total. The 2-liter return therefore contains 2 × 2/12 = 1/3 liter from A and 2 × 10/12 = 5/3 liters from B. A’s liquid remaining in B is 2 − 1/3 = 5/3 liters, equal to B’s liquid now in A. Their ratio is 1:1. The 1:5 choice instead describes the composition of B immediately before the return transfer.