Foundations test
20 questions at test level. One at a time: pick an answer and read why. At the end you see your score and the lesson to review for every miss.
In the number 7,343,915, the first 3 is worth how many times as much as the second 3?
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D. Name the places from the right: 5 is ones, 1 is tens, 9 is hundreds, the second 3 is thousands, 4 is ten thousands and the first 3 is hundred thousands. So the first 3 is worth 300,000 and the second 3 is worth 3,000. They are two places apart, and each place is ten times the one on its right: 10 × 10 = 100. Check: 3,000 × 100 = 300,000. 1 thinks the same digit always has the same value, 2 takes the two places apart as the answer instead of multiplying by ten for each one, and 10 moves only one place. The first 3 is worth 100 times as much as the second: 100.
Rounded to the nearest hundred, the headcount at a training camp is 6,300. The real headcount is a whole number. What is the difference between the largest and the smallest number it could be?
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A. Find the edges first. To round up to 6,300, the tens and ones must be 50 or more, so the smallest is 6,250 (6,249 rounds down to 6,200). To stay at 6,300, they must be less than 50, so the largest is 6,349 (6,350 rounds up to 6,400). The difference is 6,349 − 6,250 = 99. 100 uses 6,350 as the largest, 149 uses 6,200 as the smallest, and 150 makes both mistakes. The difference is 99.
A food pantry on base receives 43 boxes of canned soup with 26 cans in each box. After 275 cans are given out, how many cans are left?
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C. First the total, 43 × 26, in two rows. 43 × 6 = 258, and 43 × 20 = 860 (the zero waits at the end). 258 + 860 = 1,118 cans. Then take away the 275: 1,118 − 275 = 843. Check with an estimate: 43 × 26 is close to 40 × 25 = 1,000, so a little over 1,000 cans, minus about 300, is about 800. 69 forgets the zero in the second row (258 + 86 = 344, then 344 − 275), 543 multiplies only tens by tens and ones by ones (40 × 20 is 800, plus 18, gives 818, then 818 − 275), and 1,118 forgets to take the cans away. Cans left: 843.
Lesson: Multiplying and dividing by hand
A convoy must move 230 soldiers. Each truck carries 18 soldiers. How many trucks are needed?
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B. Long division: 18 goes into 23 once, 23 − 18 = 5, and bring down the 0 to make 50. 18 goes into 50 twice (18 × 2 = 36), and 50 − 36 = 14. So 230 ÷ 18 is 12 with a remainder of 14. Twelve full trucks leave 14 soldiers behind, and they need one more truck: 12 + 1 = 13. Check: 18 × 12 = 216, and 216 + 14 = 230. 12 drops the remainder, 14 gives the remainder as the answer, and 212 subtracts 18 instead of dividing. Trucks needed: 13.
Lesson: Multiplying and dividing by hand
At 6 a.m. the temperature at a mountain outpost is −14°F. By noon it rises 23 degrees. By midnight it falls 17 degrees from the noon reading. What is the temperature at midnight?
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C. Rising is adding and falling is subtracting. −14 + 23: the signs are different, so subtract the sizes, 23 − 14 = 9, and keep the sign of the bigger size, which is plus: 9°F at noon. Then 9 − 17: go 9 degrees down to zero and 8 more below it, so −8°F. −54°F subtracts both changes, −20°F subtracts the rise and adds the fall, and 20°F starts at 14 instead of −14. At midnight it is −8°F.
Lesson: Negative numbers
What is (−6) × (−4) ÷ (−3) − (−5)?
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B. Work left to right and let the signs decide first. (−6) × (−4): two minus signs, so plus, 24. Then 24 ÷ (−3): one minus sign, so minus, −8. Last, subtracting a negative is adding: −8 − (−5) is −8 + 5, which is −3. −13 subtracts the 5 instead of adding it (−8 − 5), 13 makes 24 ÷ (−3) positive and adds 5 (8 + 5), and 3 makes it positive and subtracts 5 (8 − 5). The answer is −3.
Lesson: Negative numbers
A supply office orders 25 cots at $48 each, plus a $75 delivery fee. What is the total cost?
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C. Times 25 is times 100, then divide by 4: 48 × 100 = 4,800, and 4,800 ÷ 4 = 1,200. Or halve first: a quarter of 48 is 12, and 12 × 100 = 1,200. Then add the fee: 1,200 + 75 = 1,275. $1,035 multiplies by 20 instead of 25 (960, plus 75), $1,200 forgets the fee, and $12,075 keeps an extra zero (12,000, plus 75). The total is $1,275.
Lesson: Mental math and estimating
A battalion has 11 companies of 136 soldiers each. 98 soldiers are on leave. How many soldiers are present?
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A. Times 11 is ten times, plus one more: 136 × 10 = 1,360, and 1,360 + 136 = 1,496. Taking away 98 is taking away 100 and giving 2 back: 1,496 − 100 = 1,396, and 1,396 + 2 = 1,398. 1,402 subtracts in columns without borrowing (in the ones it takes 6 from 8 instead of 8 from 6), 1,496 forgets the soldiers on leave, and 1,594 adds them. Soldiers present: 1,398.
Lesson: Mental math and estimating
Which fraction is the greatest?
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D. Bigger numbers do not make a bigger fraction. Compare each one with one half. 11/20 is 1/20 more than 10/20. 7/12 is 1/12 more than 6/12. 3/5 is 6/10, which is 1/10 more than 5/10. 5/8 is 1/8 more than 4/8. Eighths are the biggest of those extra pieces, so 5/8 is the greatest. Check the closest two by cross-multiplying 5/8 and 3/5: 5 × 5 = 25 and 8 × 3 = 24, and 25 is bigger, so 5/8 wins. 11/20 has the biggest numbers but the smallest extra piece. The greatest is 5/8.
A platoon of 42 soldiers has 18 women. What fraction of the platoon are men, in simplest form?
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B. The question asks about the men, so find them first: 42 − 18 = 24. Men out of the whole platoon is 24/42. Both numbers are even, so halve them: 12/21. Both divide by 3: 4/7. 3/7 is the fraction of women, 3/4 compares women to men (18/24), and 4/3 compares men to women. The men are this fraction of the platoon: 4/7.
A water can is 5/6 full at the start of a patrol and 3/8 full at the end. What fraction of the can was used?
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B. Used is start minus end: 5/6 − 3/8. The least common denominator of 6 and 8 is 24: the multiples of 8 are 8, 16, 24, and 24 is the first one that 6 goes into. 5/6 = 20/24, because 6 × 4 = 24 and 5 × 4 = 20. 3/8 = 9/24, because 8 × 3 = 24 and 3 × 3 = 9. Now subtract the tops: 20 − 9 = 11, so 11/24. 1/7 subtracts the tops but adds the bottoms (2/14), 17/24 changes the bottom of 3/8 without changing its top, and 29/24 adds the fractions instead of subtracting. Fraction of the can used: 11/24.
Lesson: Adding and subtracting fractions
A work detail paints 1/3 of a fence on Monday and 2/5 of it on Tuesday. What fraction of the fence is still unpainted?
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A. First the painted part, 1/3 + 2/5. The LCD of 3 and 5 is 15: 1/3 = 5/15 and 2/5 = 6/15, so 11/15 is painted. The whole fence is 15/15, and 15/15 − 11/15 = 4/15 is left. 11/15 is the part already painted. 3/8 adds the tops and the bottoms (1 + 2 over 3 + 5) and stops at that wrong painted part, and 5/8 takes that wrong part from the whole. The unpainted part is 4/15.
Lesson: Adding and subtracting fractions
A fuel truck holds 3,600 gallons and is 3/4 full. Then 2/3 of the fuel in it is pumped into generators. How many gallons are left in the truck?
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A. A fraction of a number: divide by the bottom, multiply by the top. 3/4 of 3,600: 3,600 ÷ 4 = 900, and 900 × 3 = 2,700 gallons in the truck. 2/3 of that is pumped out, so 1/3 stays: 2,700 ÷ 3 = 900. 2,700 is the fuel before pumping, 1,800 is the fuel pumped out, and 1,200 forgets the truck is only 3/4 full and keeps 1/3 of the whole 3,600. Gallons left in the truck: 900.
A supply clerk has 8 3/4 yards of canvas for repair patches. Each strip is 5/8 of a yard long. How many strips can the clerk cut?
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C. How many 5/8s fit into 8 3/4 is a division. First make the mixed number one fraction: 8 × 4 = 32, plus 3 is 35, so 35/4. Keep, change, flip: 35/4 × 8/5. Cancel before you multiply: 35 and 5 both divide by 5 (7 and 1), and 8 and 4 both divide by 4 (2 and 1). That leaves 7 × 2 = 14. Check: 14 strips of 5/8 is 70/8, which is 8 3/4. 5 15/32 multiplies without flipping (35/4 × 5/8 = 175/32), 1/14 flips the wrong fraction (4/35 × 5/8), and 70 multiplies by 8 but forgets to divide by 5. Strips the clerk can cut: 14.
Diesel costs $3.45 a gallon. A truck takes 18.2 gallons. What does the fill-up cost?
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B. Estimate first: about 3.5 × 18 = 63, so the answer is in the sixties. Now multiply without the points: 345 × 182. 345 × 100 = 34,500, 345 × 80 = 27,600 and 345 × 2 = 690, and 34,500 + 27,600 + 690 is 62,790. The two numbers have 2 + 1 = 3 decimal places, so put 3 places back: 62.790, which is $62.79. $6.28 and $627.90 put the point in the wrong place, and $63.70 multiplies 3.5 by 18.2, which is only the estimate. The fill-up costs $62.79.
Lesson: Decimals
Four soldiers’ times on a timed run, in minutes, are 12.08, 12.8, 12.075 and 12.48. How much faster is the fastest time than the slowest?
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D. Give every time three decimal places: 12.080, 12.800, 12.075 and 12.480. The fastest is the smallest. The tenths decide first: 0, 8, 0 and 4, so 12.800 is the slowest. 12.080 and 12.075 both have 0 tenths, so the hundredths decide, 8 against 7: 12.075 is the fastest. Line up the points and subtract: 12.800 − 12.075 = 0.725. 0.72 takes 12.08 as the fastest, 0.405 takes 12.48 as the slowest because 48 looks bigger than 8, and 0.4 makes both mistakes. The difference, in minutes, is 0.725.
Lesson: Decimals
Three recruits’ scores on a practice test are 0.805, 7/8 and 86%. What is the highest score, written as a percent?
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D. Change all three to percents. 0.805: move the point two places right, 80.5%. 7/8: one eighth is 12.5%, so seven eighths is 100% − 12.5% = 87.5%. 86% stays as it is. The highest is 87.5%. 78% writes 7/8 as 0.78, mixing up the fraction bar with a decimal point, 80.5% picks 0.805 because it has the most digits, and 86% thinks 7/8 is less than 86%. The highest score is 87.5%.
Lesson: Fractions, decimals and percents
A company of 64 soldiers puts 37.5% of them on night duty and the rest on day duty. How many are on day duty?
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D. 37.5% is a benchmark: 3/8, because one eighth is 12.5% and 3 × 12.5 = 37.5. So night duty is 3/8 of 64: 64 ÷ 8 = 8, and 8 × 3 = 24. The rest is day duty: 64 − 24 = 40. Faster: day duty is the other 5/8, and 8 × 5 = 40. 24 is the night shift, 26.5 subtracts 37.5 soldiers instead of 37.5%, and 8 stops at one eighth. On day duty: 40.
Lesson: Fractions, decimals and percents
Three generators at a field site need service: one every 6 days, one every 10 days and one every 15 days. All three were serviced today. In how many days will all three be serviced on the same day again?
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C. The day they line up again is the least common multiple of 6, 10 and 15. Count up by the biggest, 15: 15, 30. 30 ÷ 6 = 5 and 30 ÷ 10 = 3, so 30 is the first number all three go into. Check with primes: 6 is 2 × 3, 10 is 2 × 5 and 15 is 3 × 5, and the LCM takes each prime once: 2 × 3 × 5 = 30. 1 is the greatest common factor, not a multiple, 15 waits only for the longest schedule, and 900 multiplies all three numbers. They line up again in this many days: 30.
Lesson: Primes, factors, GCF and LCM
A field kitchen uses 3 quarts of milk a day. How many gallons does it use in 2 weeks?
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A. First the days: 2 weeks is 14 days, so 3 × 14 = 42 quarts. A gallon is 4 quarts, and going to a bigger unit means dividing: 42 ÷ 4 = 10.5 gallons. 42 stops at quarts, 21 divides by 2 as if a gallon were 2 quarts, and 168 multiplies by 4 instead of dividing. Gallons used: 10.5.
Lesson: Units and measurement