Ratios, Proportions and Rates

🚚 Ratios, Proportions and Rates

Part 1 of 7 — Ratios and Rates on the AR Subtest

Topic Overview

What these questions look like

Ratio and rate problems are some of the most common stories on Arithmetic Reasoning (AR). The official description of the subtest lists ratio and proportion by name, and official sample topics include rate × time and proportional rates. You will see wording like this:

• "for every 3 drivers there are 2 mechanics" (a ratio)
• "a machine fills 60 bags in 5 minutes; how many in 20 minutes?" (a proportion)
• "a truck averages 50 miles per hour" (a rate)
• "working together, how long will it take?" (a work rate)
• "which size is the better buy?" (a unit price)

Every one of these is really the same idea: two quantities that change together at a steady rate. Once you find that rate (how much of one thing per one of the other), the rest is multiplying or dividing.

The test-day reality

The CAT-ASVAB is adaptive: a right answer usually brings a harder question. Harder rate questions add a second step (convert minutes to hours, combine two workers, find an average speed for a round trip). If you take a paper version, the pace is much faster, but the skills are identical.

AR is one of the four subtests in your AFQT score (AR + MK + 2 × VE, turned into a percentile from 1 to 99), which decides whether you can enlist. AR also feeds job line scores such as the Army GT score (AR + VE).

The habits, applied to ratios and rates

1. Read the last sentence first. Is it asking for miles or hours? The smaller share or the larger one? Cost per ounce or total cost? Circle the unit it wants.
2. Translate "per" as "divide." "Miles per gallon" means miles ÷ gallons. "Price per pound" means price ÷ pounds. The word after "per" goes on the bottom.
3. Track units at every step. Put the same unit in the same position on both sides of a proportion, and turn minutes into hours before you use a speed in miles per hour.
4. Estimate first. If 4 gallons go 120 miles, then 10 gallons should go a bit more than double that distance. An estimate knocks out the choices that are wildly off.
5. Check reasonableness. More time means more distance. Two workers together finish faster than either one alone. A bigger box isn't always the cheaper buy per ounce.

The one move behind everything: find the rate for ONE

Worked Example: a delivery route

A delivery truck covers 168 miles in 4 hours. At the same speed, how far will it travel in 7 hours?

96 miles · 294 miles · 672 miles · 1,176 miles

Step 1 — Last sentence. We want a distance in miles for 7 hours.

Step 2 — Find the rate for one hour. $168 \div 4 = 42$ miles per hour.

Step 3 — Estimate. 7 hours is a little less than double 4 hours, so the answer should be a little less than $2 \times 168 = 336$ miles.

Step 4 — Compute.

$7 \times 42 = 7 \times 40 + 7 \times 2 = 280 + 14 = 294$

The truck travels 294 miles.

Step 5 — Check. 294 is a little less than 336, just as the estimate said. Here is where the other choices come from:

Key Fact: A rate problem almost always has an "upside-down" choice. Ask: should my answer be bigger or smaller than the number I started with? More hours must mean more miles.

Warm-up. Find the rate for one, then use it. 🔥

Warm-up, continued. ✍️

In a motor pool, the ratio of trucks to cars is 2 to 5. There are 14 trucks.

1) How many cars are in the motor pool? (number only, in cars)

Key Takeaways — Part 1

• Ratio, proportion and rate problems are named in the official AR description and show up often
• On the CAT-ASVAB you have about 3.5 minutes per question, no calculator, and you can't go back
• The core move: find the rate for one (per hour, per can, per minute), then multiply
• "Per" means divide, and the word after "per" goes on the bottom
• Always ask whether the answer should be bigger or smaller than what you started with; that catches the upside-down choice

Up Next: Part 2 — ratios and part-to-whole problems, including how to split a total in a given ratio.

🚚 Ratios, Proportions and Rates

Part 2 of 7 — Ratios and Part-to-Whole Problems

What a ratio says

A ratio compares two amounts. "The ratio of teens to adults is 2 to 3" can be written three ways, and they all mean the same thing:

It means: for every 2 teens there are 3 adults. It does not say there are exactly 2 teens; there could be 20 teens and 30 adults.

Order matters

The ratio is written in the same order as the words. "Teens to adults" puts teens first. "Adults to teens" is $3 : 2$, a different ratio. Flipping the order is the most common ratio mistake on AR, and the flipped ratio is almost always one of the choices.

Simplifying a ratio

Divide both numbers by the same thing, just like reducing a fraction.

Example. A volunteer group has 16 teens and 24 adults. The ratio of teens to adults is

$16 : 24 = \frac{16}{24} = \frac{2}{3}, \text{ so } 2 : 3$

(Divide both by 8.) The ratio of adults to teens is $3 : 2$.

Part-to-part vs part-to-whole

A ratio like $3 : 5$ compares one part to another part. To get a fraction of the whole, add the parts to get the total number of shares.

Example. A trail mix has peanuts and raisins in the ratio $3 : 5$.

• Total shares: $3 + 5 = 8$
• Peanuts are $\frac{3}{8}$ of the mix, and raisins are $\frac{5}{8}$ of the mix
• Peanuts are not $\frac{3}{5}$ of the mix. That fraction compares peanuts to raisins, not peanuts to everything.

In a 40-ounce bag, raisins make up $\frac{5}{8} \times 40 = 25$ ounces, and peanuts make up $40 - 25 = 15$ ounces. Check: $15 : 25 = 3 : 5$. ✓

Splitting a total in a ratio: the parts method

1. Add the parts to get the number of shares.
2. Divide the total by the number of shares to get the size of one share.
3. Multiply one share by each part.

Worked Example. A $360 bonus is split between the day shift and the night shift in the ratio $5 : 3$. How much does the day shift get?

• Shares: $5 + 3 = 8$
• One share: $360 \div 8 = 45$
• Day shift: $5 \times 45 = 225$, so $225. Night shift: $3 \times 45 = 135$, so $135.

Check: $225 + 135 = 360$ ✓, and the day shift (the bigger part) gets more than half of $360. ✓

Key Fact: The total divided by the sum of the parts gives one share. Dividing by just one of the parts is a classic wrong answer.

Three-part ratios

The parts method works the same way with three parts.

Worked Example. A concrete mix uses cement, sand and gravel in the ratio $1 : 2 : 3$. A crew needs 18 cubic feet of mix. How much gravel is in it?

• Shares: $1 + 2 + 3 = 6$
• One share: $18 \div 6 = 3$ cubic feet
• Gravel: $3 \times 3 = 9$ cubic feet (cement 3, sand 6)

Check: $3 + 6 + 9 = 18$ ✓

Changing a ratio by adding more

Sometimes a story adds more of one item and asks how many to add to reach a new ratio. Keep the unchanged amount fixed and find what the other amount must become.

Worked Example. A lot holds 10 sedans and 6 trucks. How many trucks must be added so the ratio of sedans to trucks is $5 : 4$?

• Sedans stay at 10. Since $10 = 2 \times 5$, the trucks must be $2 \times 4 = 8$.
• Trucks to add: $8 - 6 = 2$.

Check: $10 : 8 = 5 : 4$ ✓. Notice that 8 (the new total of trucks) would be a trap answer: the question asked how many to add.

Check your ratios. Watch the order of the words. 🎯

Split it with the parts method. ✍️

A painter mixes blue paint and white paint in the ratio 2 to 7 and makes 45 quarts of the mix.

1) How many quarts of blue paint are in the mix? (number only, in quarts)

2) How many quarts of white paint are in the mix? (number only, in quarts)

Ratio language. Choose the best completion for each statement. 🔽

Key Takeaways — Part 2

• A ratio is written in the same order as the words; the flipped ratio is a favorite wrong choice
• Simplify a ratio by dividing both numbers by the same amount
• Part-to-whole: in a ratio $a : b$, the first part is $\frac{a}{a + b}$ of the total
• Parts method: total ÷ (sum of the parts) = one share, then multiply by each part
• Three-part ratios work the same way; check that the pieces add back to the total
• When a ratio changes, keep the unchanged amount fixed and answer exactly what was asked (the number added, not the new total)

Up Next: Part 3 — proportions and scale: rates that stay the same, recipes and mixtures, and map and blueprint scales.

🚚 Ratios, Proportions and Rates

Part 3 of 7 — Proportions and Scale

What a proportion is

A proportion says two ratios are equal. On AR it appears whenever a rate stays the same: the same speed, the same price per item, the same recipe, the same map scale.

Set it up with matching units

Write each ratio with the same unit on top and the same unit on the bottom.

Worked Example. A press stamps 48 parts in 6 minutes. At the same rate, how many parts does it stamp in 25 minutes?

$\frac{48 \text{ parts}}{6 \text{ min}} = \frac{x \text{ parts}}{25 \text{ min}}$

Cross-multiply: if $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.

$6 \times x = 48 \times 25 = 1{,}200 \qquad x = 1{,}200 \div 6 = 200$

The unit-rate way gives the same answer: $48 \div 6 = 8$ parts per minute, and $8 \times 25 = 200$ parts.

Estimate: 25 minutes is a little more than $4 \times 6 = 24$ minutes, so the answer is a little more than $4 \times 48 = 192$. ✓

When the unknown is on the bottom

How long does the same press take to stamp 360 parts?

$\frac{48}{6} = \frac{360}{x} \qquad 48x = 6 \times 360 = 2{,}160 \qquad x = 45 \text{ minutes}$

Or with the rate: $360 \div 8 = 45$ minutes.

Key Fact: Use whichever method feels faster. The unit rate is quickest when it divides evenly; cross-multiplying always works.

Recipes and mixtures

A recipe or mixture keeps the same ratio no matter how much you make.

Worked Example: fuel-to-oil mix. A chainsaw needs 4 ounces of oil for every 2 gallons of gas. How many ounces of oil go into 5 gallons of gas?

$\frac{4 \text{ oz oil}}{2 \text{ gal gas}} = \frac{x \text{ oz oil}}{5 \text{ gal gas}} \qquad 2x = 20 \qquad x = 10 \text{ ounces}$

Unit rate: $4 \div 2 = 2$ ounces per gallon, so $5 \times 2 = 10$ ounces.

Reasonableness: 5 gallons is more gas than 2 gallons, so it needs more oil than 4 ounces. ✓

The scale-factor shortcut

When the new amount is a whole-number multiple of the old one, just multiply. A soup recipe that serves 6 uses 2 cans of broth. To serve 18, multiply everything by $18 \div 6 = 3$: you need $2 \times 3 = 6$ cans.

Watch out: Proportions multiply; they don't add. Going from 6 servings to 18 is "times 3," not "plus 12." Adding 12 cans would be way off.

Map and blueprint scale

A scale is a ratio between a drawing and real life.

Worked Example: map. The scale is 1 inch = 40 miles. Two towns are $2\frac{1}{2}$ inches apart on the map.

$2\frac{1}{2} \times 40 = 2 \times 40 + \frac{1}{2} \times 40 = 80 + 20 = 100 \text{ miles}$

Going the other way: two towns are 140 miles apart. On the map they are $140 \div 40 = 3\frac{1}{2}$ inches apart. Real distances are big numbers; map distances are small numbers. If you got 5,600 inches, the scale went the wrong way.

Worked Example: blueprint. On a blueprint, $\frac{1}{4}$ inch = 1 foot. A wall is drawn $3\frac{1}{2}$ inches long.

• 1 inch holds four quarter-inches, so 1 inch = 4 feet.
• $3\frac{1}{2} \times 4 = 12 + 2 = 14$ feet.

Key Fact: Turn an awkward scale into "1 inch = ___" first. Then drawing-to-real is multiply, and real-to-drawing is divide.

Use the map above. 🗺️

Set up the proportion. Same units in the same positions. 🎯

Read a blueprint. ✍️

On a blueprint, $\frac{1}{2}$ inch represents 3 feet. A rectangular storage room is drawn 4 inches long and $2\frac{1}{2}$ inches wide.

1) What is the actual length of the room? (number only, in feet)

2) What is the actual width of the room? (number only, in feet)

Proportion setup. 🔽

Key Takeaways — Part 3

• A proportion sets two ratios equal; put the same unit in the same position on both sides
• Cross-multiply ($a \times d = b \times c$) or use the unit rate; both give the same answer
• Recipes and mixtures multiply when they scale up; they never add
• For any scale, rewrite it as "1 inch = ___": drawing to real is multiply, real to drawing is divide
• Check direction: more time means more output, and real distances are bigger numbers than map distances

Up Next: Part 4 — distance, rate and time: $d = r \times t$, minutes to hours, two vehicles at once, and average speed.

🚚 Ratios, Proportions and Rates

Part 4 of 7 — Distance, Rate and Time

One formula, three ways to use it

Rate × time is an official AR sample topic. Everything comes from one formula:

$\text{distance} = \text{rate} \times \text{time} \qquad d = r \times t$

The units must match: miles per hour goes with time in hours.

Worked Example. A convoy drives at 52 miles per hour for 2 hours 30 minutes. How far does it go?

• 2 hours 30 minutes is $2\frac{1}{2}$ hours, or 2.5 hours.
• $52 \times 2.5 = 52 \times 2 + 52 \times 0.5 = 104 + 26 = 130$ miles.

Estimate: about 50 mph for 2.5 hours is about 125 miles. ✓

Turn minutes into hours first

A speed in miles per hour needs a time in hours. Divide the minutes by 60, and learn these by heart:

Worked Example. A truck drives 36 miles per hour for 40 minutes. How far does it go?

$40 \text{ min} = \frac{2}{3} \text{ h} \qquad 36 \times \frac{2}{3} = 12 \times 2 = 24 \text{ miles}$

The trap: $36 \times 40 = 1{,}440$ "miles." That's what you get if you forget to convert, and it fails the reasonableness test: nobody drives 1,440 miles in under an hour.

Worked Example: finding a speed. A cyclist rides 10 miles in 50 minutes. What is her speed in miles per hour?

• In 10 minutes she rides $10 \div 5 = 2$ miles.
• An hour has six 10-minute blocks: $6 \times 2 = 12$ miles per hour.

Key Fact: "Minutes left un-converted" is one of the most common AR traps. Before you multiply or divide by a speed, make sure the time is in the same unit as the speed.

Two vehicles at once

Moving apart or toward each other: ADD the speeds

Worked Example. Two buses leave the same depot at the same time and drive in opposite directions, one at 50 mph and one at 60 mph. When will they be 330 miles apart?

• Every hour the gap grows by $50 + 60 = 110$ miles.
• $330 \div 110 = 3$ hours.

The same idea works for two vehicles driving toward each other: the gap shrinks by the sum of the speeds.

One catching up to the other: SUBTRACT the speeds

Worked Example. A truck leaves a depot at 7:00 a.m. traveling 45 mph. At 8:00 a.m. a car leaves the same depot on the same road at 60 mph. When does the car catch the truck?

• Head start: by 8:00 the truck is $45 \times 1 = 45$ miles ahead.
• Closing speed: the car gains $60 - 45 = 15$ miles every hour.
• Time to catch up: $45 \div 15 = 3$ hours after 8:00, so at 11:00 a.m.

Check: at 11:00 the truck has driven 4 hours ($4 \times 45 = 180$ miles) and the car 3 hours ($3 \times 60 = 180$ miles). ✓

Average speed for a round trip

This is a hard-item favorite. Average speed = total distance ÷ total time. It is NOT the average of the two speeds.

Worked Example. A driver goes 60 miles to a training site at 30 mph and comes back the same 60 miles at 60 mph. What is the average speed for the whole trip?

$\text{average speed} = 120 \div 3 = 40 \text{ mph}$

The tempting answer is $\frac{30 + 60}{2} = 45$ mph, and it will almost certainly be a choice. It's wrong because the driver spends more time at the slow speed (2 hours) than at the fast speed (1 hour), so the average is pulled toward the slower speed.

Key Fact: For any trip with two legs, find the time for each leg, add the distances, add the times, then divide. Never average the two speeds.

Distance, rate and time. Check your units before you multiply. 🎯

Convert, then compute. ✍️

A patrol boat cruises at a steady 24 miles per hour.

1) How far does it travel in 1 hour 15 minutes? (number only, in miles)

2) How long does it take to travel 18 miles? (number only, in minutes)

Quick checks. 🔽

Key Takeaways — Part 4

• $d = r \times t$, $t = d \div r$, $r = d \div t$, with time in the same unit as the speed
• Convert minutes to hours by dividing by 60: 15 min = $\frac{1}{4}$ h, 20 min = $\frac{1}{3}$ h, 40 min = $\frac{2}{3}$ h, 45 min = $\frac{3}{4}$ h
• Moving apart or toward each other: add the speeds. One catching the other: subtract the speeds, and find the head start first
• Average speed = total distance ÷ total time, never the average of the two speeds

Up Next: Part 5 — work rates (two workers on one job) and unit prices, fuel economy and rate conversions.

🚚 Ratios, Proportions and Rates

Part 5 of 7 — Work Rates and Unit Prices

Work rates: think "jobs per hour"

If a pump drains a tank in 4 hours, then in one hour it drains $\frac{1}{4}$ of the tank. That fraction is its rate:

$\text{rate} = \frac{1 \text{ job}}{\text{time for the job}}$

When two workers (or machines) work together, their rates add, not their times.

Worked Example. One pump drains a tank in 4 hours. A second pump drains it in 12 hours. How long do they take working together?

• Rates: $\frac{1}{4}$ and $\frac{1}{12}$ of the tank per hour.
• Combined rate: $\frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$ of the tank per hour.
• Time together: $1 \div \frac{1}{3} = 3$ hours.

Shortcut for two workers: time together $= \frac{\text{product}}{\text{sum}} = \frac{4 \times 12}{4 + 12} = \frac{48}{16} = 3$ hours.

Reasonableness check: together they must be faster than the faster worker alone. 3 hours is less than 4 hours. ✓ Adding the times (16 hours) or averaging them (8 hours) gives an answer that is slower than one pump alone, which makes no sense.

Unit price and the best buy

To compare packages of different sizes, find the price per unit (per ounce, per pound, per item):

$\text{unit price} = \text{price} \div \text{quantity}$

Worked Example. A 12-ounce jar of peanut butter costs $3.00 and a 20-ounce jar costs $4.40. Which is the better buy?

The 20-ounce jar is the better buy, even though it costs more in total.

Mental-math tip: dividing by 20 is the same as dividing by 2 and then by 10: $4.40 \div 2 = 2.20$, then $2.20 \div 10 = 0.22$.

Watch out: The cheapest package is not always the best buy, and neither is the biggest one. Only the price per unit tells you.

Fuel economy and rate conversions

Miles per gallon

"Miles per gallon" (mpg) is a rate: miles ÷ gallons. To find the gallons a trip needs, divide the miles by the mpg.

Worked Example. A truck gets 18 miles per gallon. Diesel costs $3.50 per gallon. What does the fuel cost for a 252-mile trip?

1. Gallons: $252 \div 18 = 14$ gallons. (Check: $18 \times 14 = 180 + 72 = 252$.)
2. Cost: $14 \times 3.50 = 42 + 7 = 49$, so $49.00.

Estimate: about $250 \div 20 = 12.5$ gallons at about $3.50 is roughly $44, so $49 is in the right range. Stopping at 14 (the gallons) is the trap for this kind of question.

Converting a rate

Worked Example. A hose delivers 8 gallons per minute. How many gallons is that per hour, and how long does it take to fill a 1,200-gallon tank?

• Per hour: $8 \times 60 = 480$ gallons per hour.
• Time: $1{,}200 \div 8 = 150$ minutes, which is $150 \div 60 = 2\frac{1}{2}$ hours.

Key Fact: When a rate is "per minute" and the question wants hours (or the other way around), convert at the end and read the last sentence again to see which unit it wants.

Work rates and unit prices. 🎯

Convert a rate. ✍️

A fire hose pumps water at 12 gallons per minute.

1) How many gallons does it pump in one hour? (number only, in gallons)

2) How many hours does it take to fill a 2,160-gallon tank? (number only, in hours)

Rate rules. 🔽

Key Takeaways — Part 5

• A worker who takes $T$ hours has a rate of $\frac{1}{T}$ job per hour; rates add, times don't
• Time together $= 1 \div$ (combined rate), or $\frac{\text{product}}{\text{sum}}$ for two workers
• Together is always faster than the fastest worker alone
• Unit price = price ÷ quantity; the best buy has the lowest unit price
• Fuel: gallons = miles ÷ mpg, then cost = gallons × price per gallon
• Convert per-minute rates to per-hour by multiplying by 60, and answer in the unit the question asks for

Up Next: Part 6 — test strategy and the traps that rate and ratio choices are built from.

🚚 Ratios, Proportions and Rates

Part 6 of 7 — Test Strategy and Common Traps

The seven traps behind the wrong choices

Ratio and rate questions have very predictable wrong answers. Learn them and you will recognize them on sight.

Key Fact: Each trap produces a number that really is one of the four choices. Matching a choice proves nothing; a quick check against the story does.

Timing: spend your 3.5 minutes where they count

On the CAT-ASVAB you have 55 minutes for 15 scored AR questions, about 3.5 minutes each on average.

• A one-step rate (price per item, $d = r \times t$ with hours already given) should take about 1 minute.
• The time you bank pays for the multi-step ones: a catch-up problem, a round-trip average speed, two workers on one job. Those can take 4 or 5 minutes, and that's fine.
• You can't skip or go back, so give each problem real effort the first time and check before you answer.
• If you are stuck, use estimation to eliminate choices and pick the most reasonable one that's left. Unfinished questions are scored as random guesses, and a run of wrong answers late in a subtest is penalized heavily, so keep genuinely working problems instead of clicking at random. The time limit is generous enough that this rarely comes up.

Estimation eliminates choices

• Work together: the answer must be less than the faster worker's time. If one crew takes 8 hours, any choice of 8 or more is out.
• Round-trip average speed: the answer is between the two speeds and a bit below their simple average, because more time is spent at the slower speed.
• Proportions: decide first whether the answer should be bigger or smaller than the number you started with.

Back-solving: let the choices do the work

When a problem is hard to set up, plug the choices into the story.

Worked Example. One pump fills a tank in 8 hours. With a second pump helping, the tank fills in 6 hours. How long would the second pump take alone?

Choices: 12 hours, 16 hours, 24 hours, 32 hours.

Try 24 with the product-over-sum shortcut: $\frac{8 \times 24}{8 + 24} = \frac{192}{32} = 6$ hours. ✓ That's the answer.

(If you had tried 12: $\frac{8 \times 12}{20} = \frac{96}{20} = 4.8$ hours, too fast, so the second pump must be slower than 12 hours.)

The 20-second reasonableness checklist

1. Did I answer the last sentence? The number added, not the new total; the cost, not the gallons.
2. Is it in the right unit? Minutes or hours; inches on the map or miles on the ground.
3. Is the direction right? More time means more distance; a bigger recipe means more of every ingredient.
4. Do the parts add back to the whole? In a split, the shares must total the original amount.
5. Is a combined time shorter than each worker alone? Is an average speed between the two speeds?

Spot the trap. A trap answer is waiting in each one. Find the real answer. 🕵️

Spot the trap, continued. 🕵️

Key Takeaways — Part 6

• The seven traps: inverted ratio, part-to-part as part-to-whole, mismatched units, averaging speeds, adding work times, un-converted minutes, backwards map scale
• Bank time on one-step rates so you can spend 4 or 5 minutes on the multi-step ones
• Estimate to eliminate: combined time < fastest alone; average speed between the two speeds and below their simple average
• When setup is hard, back-solve with the choices
• Run the checklist: last sentence, unit, direction, parts add up, reasonableness

Up Next: Part 7 — a mixed mastery set covering every skill in this topic.

🚚 Ratios, Proportions and Rates

Part 7 of 7 — Mixed Mastery Set

Put it all together

This set mixes every skill from the topic, roughly the way a real AR subtest builds: a few quick ones, a solid middle, then a few that take more steps.

Work each one the way you would at MEPS: last sentence first, find the rate for one, units on every line, an estimate, and a final check. Aim for about 2 to 3 minutes per problem, faster on Set A.

Set A — Warm-up. 🟢

Set B — Test level. 🟡

Set C — Stretch. 🔴

One more, on scratch paper. ✍️

One sanding machine can refinish a gym floor in 4 hours. A second machine can refinish the same floor in 6 hours.

1) Working together, how many hours do the two machines take? (number only, in hours, as a decimal)

2) How many minutes is that? (number only, in minutes)

Topic Summary — Ratios, Proportions and Rates

What to do next

1. Take the exit quiz for this topic. If you score 80% or better, the topic is done.
2. Review the flashcards for this topic. They focus on the parts method, part-to-whole fractions, minutes-to-hours conversions, work rates and average speed.
3. Move on to the next AR topic: averages, counting and probability. You will use the same habits: last sentence first, units on every line, an estimate and a check.