QuanticaPrealgebraOpen in the courseA smoothie owner keeps the flavor steady by using the same mix every time, three mango chunks for every two yogurt scoops. That comparison of two quantities is a ratio, and it holds whether she is making one cup or filling a pitcher.
Problem
Each scoop drops 5 almonds and 3 dried cherries into a bag. A customer orders 4 scoops. How many dried cherries land in the bag?
Show a hint
- Every scoop is identical, and the cherries from one scoop do not depend on the almonds at all. Just count the cherries one scoop brings, then ask how many scoops there are.
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One scoop drops cherries, and there are scoops, so . The bag holds dried cherries. The almond count never enters the work, since each side of a pairing carries its own steady count.
Problem
At a kite festival the rule is 3 yellow kites for every 5 red ones. One cluster has 9 yellow kites. How many red kites are in the cluster?
Show a hint
- The phrase for every 3 yellow there are 5 red is a ratio of . The yellow part grew from to . How many times bigger is that? That same multiplier has to act on the red part too, or the comparison would change.
- To go from yellow to yellow you multiply by , since . Apply that same to the reds.
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To go from yellow to yellow you multiply by , so the reds triple too, giving . The cluster has red kites. Scaling both parts by the same number leaves the comparison alone, which is why reduces right back to .
Problem
A garden plants basil and mint in the ratio . One full repeat of the pair uses plants, so every valid total is a multiple of . What is the only possible total between 30 and 40?
Show a hint
- One full repeat of the comparison is basil and mint, so each bundle is plants. The real total has to be made of whole bundles, so it must be a multiple of .
- List the multiples of and find the one that lands between and . You have . Which single one fits?
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Each full repeat of is plants, and the multiples of are . Only sits between and , so the total is . That is bundles, basil and mint, which still compares as .
Problem
A board game ships tokens in the ratio . Reduce it to lowest terms. Write your answer as a:b.
Show a hint
- List the factors that and share. Both are even, so works, and both are divisible by , so divides each. Is there anything bigger than that divides both?
- The greatest common factor is . Divide each part by : gives the new first number and gives the new second number. Then check the two results share no common factor larger than .
Show the full solution
The greatest common factor of and is . Divide each part by it, and , so the simplest form is . Since and share no factor larger than , nothing can shrink it further. That is exactly what simplest form means.
Problem
A school library returned books on time to late in the ratio . Reduce to simplest form. Write as a:b.
Show a hint
- Simplifying a ratio works the same way as simplifying a fraction. You want the greatest common factor of and , the biggest number that divides both cleanly. Both end in a or a , so divides each, but check whether something larger does too.
- Both numbers are multiples of , since and . Divide each part of the ratio by , then check that the two results have no common factor left bigger than .
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Break each part apart, and . They share one and one , so the greatest common factor is . Dividing gives and , so the ratio is . Stopping at the easy would leave , which still reduces, so always take the greatest common factor.
Problem
A cooling fluid should compare water to coolant as . Six ratios were recorded: , , , , , . How many are equivalent to ?
Show a hint
- Test each ratio by dividing both numbers by their greatest common factor, the same move you used to reduce a fraction to lowest terms. Then see whether what is left is .
- Try . Both numbers divide by , giving , so that one matches. Now do the same simplifying step to and notice it becomes , which is not . Work through all six and keep a running count of the ones that land on .
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Reduce each one and compare with . divides by to give , a match. divides by to give , no. divides by to give , a match. divides by to give , no. divides by to give , a match. divides by to give , a match. That is matching ratios. Two ratios are equivalent exactly when they reduce to the same simplest form.
Problem
A trail map records two legs as and . Write as a whole-number ratio in lowest terms, in the form a:b.
Show a hint
- The decimals reach the hundredths place, so multiply both parts by . That turns into and into , giving the whole-number ratio . Stretching both legs by the same factor of keeps the comparison identical, the same way and are equal fractions.
- Now reduce the way you reduced fractions in Chapter 4. The largest number that divides both and is . Divide each part by and see what whole-number ratio is left.
Show the full solution
Multiply both parts by to clear the decimals. The greatest common factor of and is , so divide each part by it. The two legs compare as . Scaling both parts by the same number never changes the comparison, so clearing decimals first is always safe.
Problem
A bakery uses cup almond flour and cup tapioca starch. Write the ratio in lowest terms as a:b.
Show a hint
- Both fractions sit over a denominator that divides into . Multiply each part of the ratio by . Multiplying both parts by the same number gives an equivalent ratio, the same way and name the same fraction.
- After scaling you get . Now finish like you would simplify a fraction. Divide both parts by the largest number that goes into each.
Show the full solution
Multiply both parts by to clear the fractions. That makes the ratio , and dividing both parts by leaves . Clear the fractions first, then reduce. Trying to reduce as it stands is much harder to see.
So far a ratio has fixed only the relative sizes, never the totals. To get real amounts, read a ratio as a count of equal parts, parts of one thing and parts of the other, every part the same size. Once you know what a single part is worth, every quantity in the problem follows by multiplication.
Problem
A bag holds 56 pieces of dried fruit mixed as raisins to pumpkin seeds in the ratio . How many pumpkin seeds are in the bag?
Show a hint
- The ratio means the bag is built from equal parts of raisins and equal parts of seeds. Add those up to see how many equal parts make the whole bag.
- There are equal parts in all, and they must total pieces. So one part is . The seeds are the parts, so multiply that one part value by .
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The ratio makes equal parts, so one part is pieces. The seeds are parts, which is . The bag holds pumpkin seeds. Check it with the raisins, , and . Find what one part is worth and every amount follows.
Problem
A box of 84 stickers is split between two friends in the ratio , bigger share to the friend who paid more. How many stickers are in the smaller share?
Show a hint
- Add the two numbers in the ratio to find how many equal parts the box is divided into. Then figure out how many stickers sit in a single part.
- One part is stickers. The smaller share is the -part side, so it holds five of those parts.
Show the full solution
The ratio splits the box into equal parts, so one part is stickers. The smaller share is the -part side, which is stickers. The larger share is , and , the whole box.
Problem
A lemonade recipe uses lemon juice to water in the ratio . One batch uses 6 ounces of lemon juice. How many ounces does the whole batch hold?
Show a hint
- The lemon side of the ratio is parts, and those parts landed on six ounces. What number do you multiply by to get ? That single number is the multiplier for every part of the ratio.
- One full repeat of the recipe is parts. Multiply those parts by the multiplier you found, and you have the whole batch in ounces.
Show the full solution
The lemon juice is parts and comes to six ounces, so one part is ounces. The whole batch is parts, which is ounces. The batch holds ounces. Knowing just one quantity is enough, because it pins down the value of a part and every other amount follows.
Problem
An aquarium opens with 40 fish, blue to green in . The keeper adds only green fish until the ratio reads . How many green fish must the keeper add?
Show a hint
- The blue total never moves, it stays at . So your job is to find how much green pairs with blue when the comparison is , then compare that to the green already swimming there.
- In the new comparison , the blue is parts and equals fish, so one part is . The green is parts. Find the green count, then subtract the you started with.
Show the full solution
The opening ratio makes parts, so fish per part, giving blue and green. Only green is added, so blue stays at . In the target , blue is the side, so one part is and green must reach . That means adding green fish. The quantity that never changes is what sets the new part size, so anchor everything to the blue.
Practice these ideas
Practice
A vendor packs gift bags with 4 glitter stickers and 6 mini erasers each. She fills 5 bags. How many mini erasers does she use in total?
Show the solution
Each bag holds six mini erasers and she fills five bags, so . She uses mini erasers. The sticker count does not matter here, since the question asks about only one side of the pairing.
Practice
A swim club keeps 2 coaches for every 9 swimmers. This morning 36 swimmers arrived. How many coaches are on deck?
Show the solution
The swimmers went from to , and , so the coaches scale by as well, giving . There are coaches on deck. Both parts of a ratio move by the same multiplier, which is why reduces right back to .
Practice
A flower stand bundles tulips and daffodils in the ratio . One full repeat is stems, so every valid total is a multiple of 11. What is the only total between 70 and 80?
Show the solution
Every valid total is a multiple of . Near the range, is too small and is too big, so the total is . That is repeats, meaning tulips and daffodils.
Practice
A fabric border repeats wide stripes to narrow stripes in the ratio . Reduce first, then scale up so the wide count reaches 24. How many narrow stripes go with 24 wide ones?
Show the solution
Reduce by dividing both parts by , which gives . To get from wide stripes to you multiply by , so the narrow part becomes . That is narrow stripes. You could also scale the original by , since too. Reducing first just keeps the numbers friendlier.
Practice
A juice bar restocks mango and guava pods in the ratio . Simplify to lowest terms. Write your answer as a:b.
Show the solution
The greatest common factor of and is . Dividing gives and , so the simplified ratio is . The only factor and share is , so it is fully reduced.
Practice
A bakery sells sourdough to rye in the ratio . Simplify to lowest terms. Write your answer as a:b.
Show the solution
Both and are divisible by and by , so their greatest common factor is . Dividing gives and , so the ratio is . Now and share no factor larger than , so this is lowest terms.
Practice
A smoothie uses cup berries and cup banana. Clear the fractions and reduce to lowest terms. Write the ratio of berries to banana as a:b.
Show the solution
Both and divide evenly into , so multiply each part by . That gives and , so the ratio is . Dividing both by leaves . Any common multiple of the denominators will clear the fractions, and then you reduce what is left.
Practice
A window glaze uses cup fast-drying liquid and cup slow-drying liquid. Multiply both by 8, then reduce. Write the ratio in the form a:b.
Show the solution
Multiply both parts by . That gives and , so the ratio is . The factors of are and the factors of are , so nothing larger than divides both and it is already in lowest terms.
Practice
Two garden rows measure m and m. Rewrite as a whole-number ratio in simplest form. Write your answer as a:b.
Show the solution
Multiply both parts of by to clear the decimals, giving . Both divide by , so and , which is . Clearing decimals this way is safe because scaling both parts by the same number leaves the comparison alone.
Practice
A soup base uses cups broth and cups cream. Multiply by 10 to clear decimals, then reduce. Write the ratio as a:b.
Show the solution
Multiply both parts of by to clear the decimals, giving . The greatest common factor is , so and , leaving . That is eight parts broth for every three parts cream.
Practice
A bracelet has 72 beads, copper to jade in the ratio . How many jade beads are on the bracelet?
Show the solution
The ratio cuts the bracelet into equal parts, so one part is beads. Jade is the five-part color, which is beads. Copper gets , and , the whole bracelet.
Practice
A 90-minute playlist splits jazz to folk in the ratio . The jazz piece is the larger share. How many minutes long is the jazz piece?
Show the solution
The ratio splits the playlist into equal parts, so one part is minutes. Jazz is seven parts, which is minutes. Folk gets minutes, and , the full playlist.
Practice
A potting soil blend mixes compost to sand in the ratio . One batch uses 15 kg of compost, where one part equals one kilogram. What is the total mass of the batch?
Show the solution
The compost is the part and it weighs kilograms, so the multiplier is . One full repeat of the mix is kilograms, so the batch is kilograms. Check by parts, kilograms of compost and of sand, and .
Practice
A reef tank holds 35 fish in the ratio blue to gold. The keeper adds only blue fish until the count is . How many blue fish must she add?
Show the solution
The ratio makes equal parts, so one part is fish, giving blue and gold. Only blue is added, so gold stays locked at , and means blue has to climb to as well. It started at , so she adds blue fish. Anchor to the count that does not move.