Lesson Test: Whole Numbers and Order of Operations

Answer: A

Correct answer: 31. Multiplication comes before addition:

• A (correct): Multiply first, 6 x 4 = 24, then add 7 to get 31.
• B: This adds 7 and 6 first, but multiplication comes before addition.
• C: This multiplies all three numbers, ignoring the addition sign.
• D: This adds all three numbers, ignoring the multiplication sign.

Answer: C

Correct answer: 1,257. Count up to avoid borrowing across zeros: 1,748 to 1,800 is 52, and 1,800 to 3,005 is 1,205. So the difference is

Check:

• A: This adds the two numbers instead of subtracting.
• B: This subtracts the smaller digit from the larger one in every column instead of borrowing.
• C (correct): Counting up from 1,748 to 1,800 is 52 and from 1,800 to 3,005 is 1,205, for 1,257 in all.
• D: This is a borrowing slip across the zeros; 1,267 + 1,748 gives 3,015, not 3,005.

Answer: A

Correct answer: 5,814. A number is divisible by 9 when its digit sum is a multiple of 9. For 5,814 the sum is

• A (correct): Its digits add to 5 + 8 + 1 + 4 = 18, a multiple of 9, so it divides evenly by 9.
• B: Its digits add to 16, which is not a multiple of 9.
• C: Its digits add to 20, which is not a multiple of 9.
• D: Its digits add to 17, which is not a multiple of 9.

Answer: D

Correct answer: 4 buses. Divide: $150÷40$ is 3 remainder 30, because

The question asks how many buses are needed, so the 30 leftover recruits require a fourth bus.

• A: Three buses seat only 120 recruits, leaving 30 recruits without a seat.
• B: A bus count must be a whole number, so a decimal quotient cannot be the answer here.
• C: This reports the remainder, the number of extra recruits, as the number of buses.
• D (correct): 150 / 40 is 3 remainder 30, and the 30 extra recruits need one more bus, so 4 buses.

Answer: A

Correct answer: 4,653. Since 99 is 100 minus 1, compute

and subtract 47 once:

• A (correct): Multiply by 100 to get 4,700, then subtract 47 once: 4,700 - 47 = 4,653.
• B: This subtracts 47 twice; only one extra 47 is included when multiplying by 100.
• C: This adds 47 to 4,700, but 99 is one less than 100, so 47 must be subtracted.
• D: This multiplies by 100 and forgets to take away the extra 47.

Answer: C

Correct answer: 28. Do multiplication and division from left to right:

Then add:

• A: This adds 2 + 3 first as if they were grouped, giving 8 x 5 x 4.
• B: This multiplies 6 x 2 before dividing, but division and multiplication are done left to right.
• C (correct): Left to right, 48 / 6 = 8 and 8 x 2 = 16; then 3 x 4 = 12, and 16 + 12 = 28.
• D: This works strictly left to right, adding 3 before multiplying by 4.

Answer: A

Correct answer: 30. Work from the inside out:

Inside the brackets,

• A (correct): Innermost 9 - 6 = 3, then 4 + 2 x 3 = 10, then 3 x 10 = 30, and 60 - 30 = 30.
• B: This adds 4 + 2 before multiplying by 3 inside the brackets, giving 60 - 54.
• C: This drops the 2 in front of the parentheses, using 4 + 3 = 7 inside the brackets.
• D: This subtracts 60 - 3 first and multiplies by the bracket, but the 3 multiplies the bracket before subtracting.

Answer: B

Correct answer: 32 bolts.

so $500÷36$ is 13 remainder 32. The remainder is the number of bolts in the partial bag. Check:

• A: This is the number of bags needed in all, not a count of bolts.
• B (correct): 500 / 36 is 13 remainder 32, so 13 bags are full and the last bag holds 32 bolts.
• C: This is how many more bolts would be needed to fill the last bag.
• D: This is the number of full bags, not the number of bolts in the partial bag.

Answer: B

Correct answer: 29. Parentheses first:

Exponent:

Multiply:

Add:

• A: This ignores the parentheses and computes 5 + 8 x 7 - 4.
• B (correct): Parentheses give 3, the exponent gives 8, then 8 x 3 = 24 and 5 + 24 = 29.
• C: This treats 2 cubed as 2 x 3 = 6 instead of 2 x 2 x 2 = 8.
• D: This adds 5 and 8 before multiplying, but multiplication comes before addition.

Answer: C

Correct answer: 200. Round to friendly numbers:

Check with the inverse:

close to 7,912.

• A: This is ten times too small; 39 x 20 is only 780.
• B: This is a thousand times too large; dividing by 39 must give far less than 7,912.
• C (correct): Rounding gives 8,000 / 40 = 200, and the exact quotient is a little under 203.
• D: This is ten times too large; 39 x 2,000 is about 78,000.

Answer: B

Correct answer: 1,925 liters. Fuel drawn:

After fueling:

After the delivery:

• A: This subtracts the delivery instead of adding it.
• B (correct): The trucks draw 15 x 65 = 975 liters, 2,400 - 975 = 1,425, and 1,425 + 500 = 1,925.
• C: This adds the fuel drawn and subtracts the delivery, reversing both steps.
• D: This stops after the trucks fuel up and leaves out the 500-liter delivery.

Answer: D

Correct answer: 16. Parentheses:

Exponent:

Left to right:

Subtract:

• A: This multiplies 6 x 3 before dividing; division and multiplication must be done left to right.
• B: This treats 2 squared as 2, so only 2 x 5 = 10 is subtracted from 36.
• C: This subtracts 36 - 4 before multiplying by 5, but multiplication comes before subtraction.
• D (correct): Parentheses give 6, 72 / 6 x 3 = 36, 2 squared is 4 and 4 x 5 = 20, so 36 - 20 = 16.

Answer: D

Correct answer: 5 trucks. Boxes: $1,000÷12$ is 83 remainder 4, so 84 boxes are needed. Trucks: $84÷20$ is 4 remainder 4, so a fifth truck is needed for the last 4 boxes. Answer: 5 trucks.

• A: Four trucks carry only 80 boxes, but 84 boxes must be moved.
• B: This is the number of boxes needed, not the number of trucks.
• C: This divides the 1,000 meal packs by 20, treating each truck as holding 20 packs instead of 20 boxes.
• D (correct): 1,000 / 12 is 83 remainder 4, so 84 boxes are needed, and 84 / 20 is 4 remainder 4, which needs 5 trucks.

Answer: B

Correct answer: 4,326. A number is divisible by 6 only if it passes both the 2 test and the 3 test. 4,326 ends in 6 (even) and has digit sum

a multiple of 3. Indeed,

• A: Its digits add to 15, so it is divisible by 3, but it is odd, so it fails the test for 2.
• B (correct): It is even, and its digits add to 15, a multiple of 3, so it passes both tests for 6.
• C: It is even, but its digits add to 13, which is not a multiple of 3.
• D: It ends in an odd digit, so it fails the test for 2 and cannot be divisible by 6.

Answer: D

Correct answer: $216. Cost:

Sales:

for 1,056 dollars. Profit:

• A: This counts only the 18 kits sold at $52 and leaves out the 6 kits sold at $20.
• B: This is the total money taken in from sales; the $840 cost still has to be subtracted.
• C: This is the profit on the 18 higher-priced kits and ignores the $90 lost on the other 6.
• D (correct): Sales bring in 18 x 52 + 6 x 20 = 1,056 dollars, and 1,056 - 840 = 216 dollars of profit.

Answer: C

Correct answer: 12 trucks. Divide:

so 214 ÷ 18 is 11 remainder 16.

Eleven trucks are full, but the 16 remaining soldiers still need a ride, so round up to 12 trucks.

• A: 11 is the number of completely full trucks; it leaves 16 soldiers without a ride.
• B: 16 is the remainder, the number of soldiers left over, not a count of trucks.
• C (correct): 214 ÷ 18 is 11 remainder 16, and the 16 extra soldiers need one more truck, so 12 trucks are required.
• D: Only one extra truck is needed for the 16 leftover soldiers, so 13 is one truck too many.

Answer: C

Correct answer: 27. Multiplication and division are done in the order they appear, from left to right.

First

• A: This comes from multiplying 4 × 3 first and then dividing 36 by 12, which breaks the left-to-right rule.
• B: This divides 36 by 3 and ignores the 4, mixing up which numbers go with each operation.
• C (correct): Division and multiplication share a level, so work left to right: 36 ÷ 4 = 9, then 9 × 3 = 27.
• D: This adds the three numbers instead of dividing and multiplying.

Answer: A

Correct answer: 1,200. Use the friendly factor 25: multiplying by 25 is the same as multiplying by 100 and dividing by 4.

• A (correct): Since 4 × 25 = 100, divide 48 by 4 to get 12 and multiply by 100 to get 1,200.
• B: This drops a zero; 25 × 48 must be more than 25 × 40 = 1,000.
• C: This adds an extra zero; the product is far smaller than 25 × 500.
• D: This misplaces digits; a correct product of 25 × 48 ends in 00 because 48 is a multiple of 4.

Answer: D

Correct answer: $49. Divide the total pay by the hours: 1,176 ÷ 24. Since

which is 24 too many, the quotient is

Check with the inverse:

• A: 48 × 24 = 1,152, which is $24 short of the total paid.
• B: 59 × 24 = 1,416, which is more than the total; a digit slipped during the division.
• C: 47 × 24 = 1,128, which falls $48 short of the total paid.
• D (correct): 1,176 ÷ 24 = 49, and the check 49 × 24 = 1,176 confirms it.

Answer: C

Correct answer: 32. Division and multiplication come first:

Then add and subtract from left to right:

• A: This works strictly left to right through every operation, ignoring that division and multiplication come first.
• B: This adds 6 + 8 before subtracting, but addition and subtraction must be done left to right.
• C (correct): Do the division and multiplication first (6 and 8), then work left to right: 30 − 6 + 8 = 32.
• D: This adds the 6 instead of subtracting it.

Answer: A

Correct answer: 38. Exponent first: 5² = 25. Then multiplication and division:

Finally subtract:

• A (correct): Square first (25), then 2 × 25 = 50 and 24 ÷ 2 = 12, so 50 − 12 = 38.
• B: This subtracts 50 − 24 before dividing, but division comes before subtraction.
• C: This squares 2 × 5 = 10 to get 100; the exponent applies only to the 5.
• D: This treats 5² as 5 × 2 = 10 instead of 5 × 5 = 25.

Answer: A

Correct answer: $430. To multiply by 5, multiply by 10 and take half.

so the total is $430.

• A (correct): Multiply by 10 and halve: 86 × 10 = 860, and 860 ÷ 2 = 430.
• B: This multiplies by 10 but forgets to halve the result.
• C: This is ten times too large; 86 boxes at $5 must cost less than 100 × $5 = $500.
• D: This comes from a carrying slip; 80 × 5 = 400 and 6 × 5 = 30 give 430.

Answer: D

Correct answer: 12. Work from the innermost grouping outward. Parentheses:

Brackets:

Inside the braces:

Finally, the 3 next to the braces means multiply:

• A: This subtracts 20 − 2 before multiplying 2 by the brackets, giving 18 × 8 × 3.
• B: This drops the 2 in front of the brackets, giving 3 × (20 − 8).
• C: This multiplies only the 20 by 3, giving 60 − 16, instead of multiplying the whole braced value by 3.
• D (correct): Inside out: 12 − 9 = 3, 5 + 3 = 8, 2 × 8 = 16, 20 − 16 = 4, and 3 × 4 = 12.

Answer: A

Correct answer: 5,316. Test 3 with the digit sum and test 4 with the last two digits.

For 5,316:

a multiple of 3, and 16 is a multiple of 4, so it passes both tests.

• A (correct): Its digits add to 15, a multiple of 3, and its last two digits, 16, form a multiple of 4.
• B: Its digit sum, 15, passes the 3 test, but its last two digits, 18, are not a multiple of 4.
• C: Its last two digits, 24, pass the 4 test, but its digit sum, 16, is not a multiple of 3.
• D: Its digit sum is 10 and its last two digits are 30, so it fails both tests.

Answer: D

Correct answer: 20,000. Round each factor to a friendly number: 52 is about 50 and 398 is about 400.

which separates the choices easily. The exact product, 20,696, confirms it.

• A: This is ten times too small; 50 crates of about 400 is far more than 2,000.
• B: This is ten times too large, usually from adding an extra zero while rounding.
• C: This is about the size of a single crate, not 52 crates.
• D (correct): Rounding gives 50 × 400 = 20,000; the exact total, 20,696, is closest to this choice.

Answer: B

Correct answer: 73 remainder 4.

leaving 55, and

fits into 55 with 4 left. So the quotient is 73 remainder 4.

Check: multiply the quotient by the divisor and add the remainder:

• A: A remainder of 21 is larger than the divisor 17, so one more 17 fits; the quotient is too small.
• B (correct): 73 × 17 = 1,241, and 1,241 + 4 = 1,245, so the check returns the original number.
• C: 74 × 17 = 1,258, which is already more than 1,245, so 74 is too large.
• D: 73 × 17 + 14 = 1,255, not 1,245, so the remainder is wrong.

Answer: C

Correct answer: 883. Use compensation: 497 is 3 less than 500.

• A: This drops a carry in the tens column.
• B: This drops the carry into the hundreds column; the sum must be more than 500 + 300 = 800.
• C (correct): Add 500 instead of 497 to get 886, then take back the extra 3 to get 883.
• D: This adds the extra 3 back instead of subtracting it when using compensation.

Answer: A

Correct answer: 2,534 bottles. Bottles issued:

Bottles received:

Start with 2,750, subtract what left, and add what arrived:

• A (correct): Issued 18 × 24 = 432 and received 9 × 24 = 216, so 2,750 − 432 + 216 = 2,534.
• B: This adds the issued bottles instead of subtracting them.
• C: This subtracts the received bottles instead of adding them.
• D: This adds 9 bottles instead of 9 cases of 24 bottles after the issue.

Answer: C

Correct answer: 48. The sum of 9 and 6 is multiplied, so it goes in parentheses, and the quotient of 36 and 3 is a separate term: (9 + 6) × 4 − 36 ÷ 3.

Parentheses: 15. Then

Finally

• A: This leaves out the parentheses, so only the 6 is multiplied by 4: 9 + 24 − 12 = 21.
• B: This subtracts 36 from 60 and then divides by 3, but only 36 is to be divided by 3.
• C (correct): The sum needs parentheses: (9 + 6) × 4 − 36 ÷ 3 = 60 − 12 = 48.
• D: This subtracts 36 itself instead of the quotient of 36 and 3.

Answer: B

Correct answer: $1,900. To multiply by 25, multiply by 100 and divide by 4.

so the vests cost $1,900.

• A: This comes from a slip in the partial products: 70 × 25 = 1,750 and 6 × 25 = 150, not 100.
• B (correct): Multiply by 100 and divide by 4: 7,600 ÷ 4 = 1,900.
• C: This drops a zero; 76 vests at $25 must cost more than 40 × $25 = $1,000.
• D: This adds a zero; the product is less than 100 × $25 = $2,500.

Answer: B

Correct answer: 2 vans. Divide by the bus size:

so 9 buses are full and 16 soldiers are left over.

Those 16 soldiers need vans: 16 ÷ 12 is 1 remainder 4, so 2 vans are needed.

• A: One van seats only 12, so 4 of the 16 leftover soldiers would have no seat.
• B (correct): 9 full buses carry 396, leaving 16 soldiers, and 16 soldiers need 2 vans of 12.
• C: 16 is the number of leftover soldiers, not the number of vans.
• D: 10 is the number of buses that would be needed for everyone, not the number of vans.

Answer: D

Correct answer: 737. To multiply a two-digit number by 11, put the sum of its digits between them. Here

so write 3 in the middle and carry 1 into the 6.

Check:

• A: This writes the 3 from 13 in the middle but forgets to carry the 1 into the left digit.
• B: This places all of 13 between the digits instead of carrying the 1.
• C: This puts the wrong middle digit; 67 × 11 = 670 + 67 = 737.
• D (correct): Place the digit sum 6 + 7 = 13 between the digits and carry 1 into the 6: 737.

Answer: C

Correct answer: 5,742. To multiply by 99, multiply by 100 and subtract the number once.

• A: This adds 58 to 5,800 instead of subtracting it.
• B: This subtracts 158 instead of 58 from 5,800.
• C (correct): 58 × 100 = 5,800, and subtracting one 58 gives 5,742.
• D: This is 1,000 too small, from a borrowing slip in the thousands place.

Answer: D

Correct answer: 810. Halving one factor and doubling the other keeps the product the same.

• A: This is 40 × 18, which leaves out the cups on 5 of the tables.
• B: This is 45 × 20, which rounds 18 up and never takes back the extra cups.
• C: This adds an extra zero; 45 × 18 is less than 50 × 20 = 1,000.
• D (correct): Halve 18 and double 45: 9 × 90 = 810.

Answer: C

Correct answer: 1,692. Break 36 into 30 and 6 and multiply each part by 47:

Add the partial products:

An estimate of

shows the answer is a little under 1,800.

• A: This forgets to shift the tens partial product, adding 141 instead of 1,410 to 282.
• B: This drops a carry when adding the partial products.
• C (correct): 47 × 30 = 1,410 and 47 × 6 = 282, and 1,410 + 282 = 1,692.
• D: This adds an extra 100 while combining the partial products.

Answer: B

Correct answer: 11 eggs. Divide: 95 ÷ 12 is 7 remainder 11, because

The question asks for the leftover amount, so the last carton holds 11 eggs.

• A: 7 is the number of full cartons, not the number of leftover eggs.
• B (correct): 12 × 7 = 84, and 95 − 84 = 11 eggs are left for the last carton.
• C: 8 is the total number of cartons used, not the number of eggs in the last one.
• D: 12 is what a full carton holds; the last carton is only partly filled, so it holds fewer.

Answer: D

Correct answer: 3,516. A number is divisible by 4 when its last two digits form a multiple of 4.

Only 3,516 passes, because

Being even is not enough.

• A: It is even, but its last two digits, 38, are not a multiple of 4.
• B: It ends in 0, but its last two digits, 10, are not a multiple of 4.
• C: It is even, but its last two digits, 22, are not a multiple of 4.
• D (correct): Its last two digits, 16, form a multiple of 4.

Answer: A

Correct answer: 53. Parentheses first:

Exponent next: 2² = 4.

Then divide and multiply from left to right:

Finally,

• A (correct): Parentheses give 8, the exponent gives 4, then 96 ÷ 8 = 12, 12 × 4 = 48, and 48 + 5 = 53.
• B: This multiplies 8 × 4 before dividing, giving 96 ÷ 32 = 3, which breaks the left-to-right rule.
• C: This ignores the parentheses, computing 96 ÷ 2 + 6 × 4 + 5.
• D: This squares 12 × 2 instead of squaring only the 2.

Answer: A

Correct answer: 23 manuals. Divide:

so 1,150 ÷ 48 is 23 remainder 46.

The question asks for complete manuals, so the leftover 46 sheets are dropped, giving 23 manuals.

• A (correct): 48 × 23 = 1,104, and the 46 sheets left are not enough for another manual.
• B: Rounding up does not work here: 48 × 24 = 1,152 needs 2 more sheets than the shop has.
• C: 46 is the number of sheets left over, not a count of manuals.
• D: 22 manuals use only 1,056 sheets, leaving enough paper for one more.

Answer: C

Correct answer: 36. Work inside the parentheses first:

Then multiply:

• A: This ignores the parentheses and multiplies 3 × 8 before adding 4, counting the empty canteens only once.
• B: This adds all three numbers instead of multiplying the 3 by the sum.
• C (correct): Parentheses first: 8 + 4 = 12, then 3 × 12 = 36.
• D: This multiplies all three numbers, 3 × 8 × 4, instead of adding 8 and 4.

Answer: C

Correct answer: 5. A number is divisible by 9 when its digits add to a multiple of 9. The known digits add to

The next multiple of 9 is 18, so the missing digit is

giving

• A: The digit sum would be 17, which is not a multiple of 9.
• B: The digit sum would be 16, which is not a multiple of 9.
• C (correct): The digits then add to 4 + 7 + 5 + 2 = 18, a multiple of 9.
• D: The digit sum would be 13, which is not a multiple of 9.

Answer: B

Correct answer: $24. Tarps:

Rope:

Total cost:

Change:

so the crew receives $24.

• A: This subtracts only the cost of the tarps and forgets the rope.
• B (correct): Tarps cost $322 and rope costs $54, for $376 in all, so the change is $400 − $376 = $24.
• C: This subtracts the tarps and the price of just one coil of rope.
• D: $376 is the total cost of the purchase, not the change.

Answer: D

Correct answer: 2,725 miles. Count up from 38,475 to 41,200 in friendly steps: 25 to reach 38,500, 500 to reach 39,000, and 2,200 to reach 41,200.

Add the steps:

Check:

• A: This forgets to borrow from the ten-thousands and thousands places, adding an extra 1,000.
• B: This makes a borrowing slip across the zeros, adding an extra 100.
• C: This miscounts the first step up to 38,500, which is 25, not 35.
• D (correct): Count up: 38,475 to 38,500 is 25, to 39,000 is 500, and to 41,200 is 2,200; 25 + 500 + 2,200 = 2,725.

Answer: B

Correct answer: 52. Multiplication and division come before subtraction, and divisions are done left to right:

Then

• A: This divides 6 ÷ 2 first and then 48 ÷ 3 = 16, which breaks the left-to-right rule.
• B (correct): 7 × 8 = 56, then 48 ÷ 6 = 8 and 8 ÷ 2 = 4 from left to right, so 56 − 4 = 52.
• C: This stops after 48 ÷ 6 = 8 and never divides by 2.
• D: This adds the 4 instead of subtracting it.

Answer: D

Correct answer: 37 packs. First find the total batteries:

Then divide by the pack size:

so 870 ÷ 24 is 36 remainder 6. The extra 6 batteries require one more pack, so 37 packs are needed.

• A: 36 packs hold only 864 batteries, which leaves the issue 6 batteries short.
• B: 6 is the remainder, the number of batteries needed from the last pack, not the number of packs.
• C: This divides 145 soldiers by 24 and forgets that each soldier needs 6 batteries.
• D (correct): 145 × 6 = 870, and 870 ÷ 24 is 36 remainder 6, so a 37th pack is needed for the last 6 batteries.

Answer: D

Correct answer: 2,451. A number is divisible by 3 when the sum of its digits is a multiple of 3.

For 2,451:

so 2,451 is divisible by 3.

• A: Its digits add to 19, which is not a multiple of 3.
• B: Its digits add to 8, which is not a multiple of 3.
• C: Its digits add to 10, which is not a multiple of 3; ending in 3 does not matter.
• D (correct): Its digits add to 2 + 4 + 5 + 1 = 12, a multiple of 3.

Answer: B

Correct answer: 8,000 miles. Round to friendly numbers: 412 is about 400 and 19 is about 20.

The choices are far apart, so the estimate is enough; the exact product is 7,828.

• A: This is ten times too small; two weeks of driving alone is more than 800 miles.
• B (correct): Rounding gives 400 × 20 = 8,000; the exact total, 7,828 miles, is closest to this value.
• C: This is ten times too large, from adding an extra zero while rounding.
• D: This is a hundred times too large for 19 weeks of driving.

Answer: A

Correct answer: 208. 36 goes into 74 twice (72), leaving 2. Bring down the 8: 36 does not go into 28, so write 0 in the tens place. Bring down the last 8: 36 goes into 288 exactly 8 times.

The quotient is 208. Check:

An estimate,

rules out 28 and 2,080.

• A (correct): 36 × 200 = 7,200, the remaining 288 is 36 × 8, so the quotient is 208.
• B: This skips the zero in the tens place; 36 × 28 = 1,008, far less than 7,488.
• C: This puts an extra zero in the quotient; 36 × 2,080 is about 75,000.
• D: 36 × 218 = 7,848, which is more than 7,488, so a digit is wrong.

Answer: A

Correct answer: 428. Use the inverse operation: number = quotient × divisor + remainder.

Check: 428 ÷ 9 is 47 with 5 left over.

• A (correct): Multiply the quotient by the divisor and add the remainder: 9 × 47 + 5 = 423 + 5 = 428.
• B: This multiplies 9 × 47 but forgets to add the remainder.
• C: This subtracts the remainder instead of adding it.
• D: This is 9 × 48, which would give a quotient of 48 with no remainder.

Answer: B

Correct answer: $52. Gallons used:

since

is 32 too many.

Total cost:

Each share:

so each friend pays $52.

• A: $156 is the total fuel cost before it is split among the three friends.
• B (correct): 1,248 ÷ 32 = 39 gallons, 39 × $4 = $156, and $156 ÷ 3 = $52.
• C: This splits the 39 gallons three ways but never multiplies by the $4 price.
• D: This multiplies by 3 instead of dividing by 3 at the last step.