Alpha School · Math · built 2 October 2026
This guide covers the nine kinds of question a Grade 5 math test asks, in the order the skills build on each other. Each section starts with something you can already do, adds one new move, gives you the rule in a sentence, and names the mistake to watch for. Then you practise.
Every question here is new. None of them is from a test. The numbers and stories are different on purpose, so what you are learning is the method, not the answer.
How to use it: answer a question, press Check, then read the answer even when you were right. The answer tells you where each wrong option comes from, and that is the part worth remembering. Your work is saved in this browser, so you can close the page and come back.
In this guide
Which is more, 7 tenths or 7 hundredths?
7 tenths is 0.7. Written in hundredths it is 0.70, which is 70 hundredths.
7 hundredths is 0.07, which is only 7 hundredths.
So 0.7 is greater than 0.07. The digit 7 is the same. The place it sits in is what matters.
Now compare 0.308 and 0.38. Give every number the same number of digits by adding zeros on the right.
0.308 stays 0.308 (308 thousandths).
0.38 becomes 0.380 (380 thousandths).
380 is more than 308, so 0.38 is greater than 0.308.
Rounding uses the same idea. To round 4.2371 to the nearest hundredth, find the hundredths digit (3) and look at the one digit to its right (7). 7 is 5 or more, so the 3 rounds up.
4.2371 rounded to the nearest hundredth is 4.24.
When a question asks which list is not in order, check every neighbouring pair in each list. One pair out of order is enough.
Line up the decimal points and add zeros until every number has the same number of digits. Then compare the way you compare whole numbers.
To round, find the place, look one digit to the right. 5 or more rounds up.
Longer looks bigger. 0.308 has more digits than 0.38, so it feels bigger. With whole numbers, more digits does mean bigger. With decimals it does not. Add the zeros so both numbers have the same length, then compare.
When rounding, look at one digit to the right, not the whole tail. 6.8349 rounds to 6.83 even though the tail ends in a 9.
Practice
Write 0.6 as hundredths.
0.60
6 tenths is 60 hundredths. Adding a zero on the right does not change the value.
Which symbol makes this comparison true?
2.45 ○ 2.405
2.45 > 2.405
Write 2.45 as 2.450. Then 450 thousandths is more than 405 thousandths.
Choosing "less than" comes from seeing that 2.405 has more digits. Choosing "equal" comes from ignoring the 5 in 2.405.
Round 6.8349 to the nearest hundredth.
6.83
The hundredths digit is 3. The digit to its right is 4. 4 is less than 5, so the 3 stays.
6.84 comes from looking at the 9 at the end instead of the 4 next door. 6.8 is rounding to tenths. 6.835 is rounding to thousandths.
Which list is NOT in order from least to greatest?
0.72, 0.8, 0.718, 0.9
Check each neighbouring pair. 0.8 is 0.800 and 0.718 is 0.718. 800 is more than 718, so 0.8 then 0.718 goes down, not up.
The other three lists climb at every step. The list 3.4, 3.42, 3.5, 3.55 is the one that tempts people, because 3.4 looks shorter than 3.42, but 3.40 is less than 3.42, so it is in order.
What is 325 ÷ 5?
5 goes into 32 six times with 2 left. 5 goes into 25 five times.
325 ÷ 5 = 65.
And what is 3.25 + 1.4?
Line up the decimal points. 1.4 becomes 1.40.
3.25 + 1.40 = 4.65.
Dividing money works like the 325 ÷ 5 you just did. Imani saved the same amount each week for 6 weeks and has $58.50. How much did she save each week?
Divide 58.50 by 6 as if it were 5850 ÷ 6.
5850 ÷ 6 = 975.
Put the decimal point back, two places from the right.
58.50 ÷ 6 = 9.75. She saved $9.75 each week.
Multiplying decimals: multiply as whole numbers, then count the decimal places in both factors.
0.6 × 0.4. First 6 × 4 = 24.
0.6 has one decimal place and 0.4 has one. That is two places in total.
0.6 × 0.4 = 0.24.
Estimating "about how many times as long": round both numbers to friendly ones, then divide. 23.8 hours compared with 5.9 hours is about 24 ÷ 6, which is about 4 times as long.
Add and subtract: line up the decimal points.
Multiply: multiply as whole numbers, then count the decimal places in both factors.
Divide by a whole number: divide as whole numbers, then put the decimal point back in the same place.
0.6 × 0.4 is not 2.4. Each factor has one decimal place, so the answer needs two. Moving the point only once is the most common slip, because the factors each look like "one place" numbers.
In division, a total that does not share out evenly is a sign to add a zero and keep dividing, not a sign to stop.
Practice
What is 7.5 − 2.75?
4.75
Write 7.5 as 7.50 and line up the points.
7.50 − 2.75 = 4.75.
5.25 comes from subtracting the smaller digit from the bigger digit in each column instead of regrouping. 5.75 comes from lining up the right-hand digits instead of the decimal points.
Marisol paid $47.25 for 9 identical notebooks. Which expression gives the cost of one notebook?
47.25 ÷ 9
A total shared equally among 9 is a division, total first.
47.25 × 9 would be the cost of 9 more sets. 9 ÷ 47.25 has the numbers the wrong way round. 47.25 − 9 takes away 9 dollars, not 9 notebooks.
What is 0.7 × 0.3?
0.21
7 × 3 = 21.
One decimal place plus one decimal place is two places.
0.7 × 0.3 = 0.21.
2.1 moves the point once. 21 forgets the point altogether.
Darius saved the same amount each week for 8 weeks. He now has $94.80. How much did he save each week?
$11.85
Treat 94.80 as 9480 and divide.
9480 ÷ 8 = 1185.
Put the point back two places from the right. 94.80 ÷ 8 = 11.85.
$1.185 and $118.50 both come from misplacing the point. Check by estimating: 8 × 12 is 96, so the answer should be a little under 12.
Mixed check 1
These two questions do not say which skill they use. Decide the method first, then answer.
Keiko paid $38.40 for 6 identical plant pots. How much did one pot cost?
Method: Decimal operations
$6.40
A total shared equally is a division. Treat 38.40 as 3840.
3840 ÷ 6 = 640.
Put the point back two places. 38.40 ÷ 6 = 6.40.
Nothing here needs comparing or rounding; the words "paid" and "identical" tell you to share a total.
A bolt measures 1.608 centimeters. What is this length rounded to the nearest hundredth?
Method: Compare and round decimals
1.61
Hundredths digit is 0. The digit to its right is 8. 8 is 5 or more, so the 0 becomes 1.
1.60 forgets to round up. 1.6 is rounding to tenths. No arithmetic with the number is needed, which is how you know it is a rounding question.
How many 20s are in 160?
20 × 8 = 160, so there are 8 twenties in 160.
160 ÷ 20 = 8.
A two-digit divisor like 24 is the same idea with an estimate to get you started. Find 1,392 ÷ 24.
Estimate: 24 is close to 25, and 25 × 50 = 1,250. The answer is a bit more than 50.
24 goes into 139 five times. 24 × 5 = 120. 139 − 120 = 19.
Bring down the 2 to make 192. 24 × 8 = 192.
1,392 ÷ 24 = 58.
Now a story: 275 students sit in rows of 24. How many rows are needed?
275 ÷ 24 is 11 with 11 left over.
Eleven full rows seat 264 students. The other 11 students still need a row.
12 rows are needed.
"Times as much" is multiplication. A puppy weighs 14 pounds and its mother weighs 6 times as much. The mother weighs 14 × 6, which is 84 pounds.
Estimate first with a friendly divisor, then divide.
When the question asks how many are needed, a remainder means one more.
Dropping the remainder. 275 ÷ 24 gives 11 in the quotient, so 11 feels like the answer. But 11 rows leave people standing. Read the question again after you divide: does the leftover need a place?
"Times as much" is not "more than". 6 times as much means multiply by 6, not add 6.
Practice
What is 36 × 14?
504
36 × 10 = 360.
36 × 4 = 144.
360 + 144 = 504.
You need to find 1,806 ÷ 42.
Estimate: 1,800 ÷ 40, about 45. Exact answer: 43
Round both numbers to friendly ones. 1,800 ÷ 40 = 45, so the answer is near 45.
42 goes into 180 four times. 42 × 4 = 168. 180 − 168 = 12.
Bring down the 6 to make 126. 42 × 3 = 126.
1,806 ÷ 42 = 43.
The other estimates drop a zero from one number but not the other, or multiply instead of divide.
A hall has 19 seats in each row. How many rows are needed for 450 people to each have a seat?
24 rows
450 ÷ 19 is 23 with 13 left over.
23 rows seat 437 people. The other 13 people need one more row.
24 rows are needed.
23 drops the remainder. 23 R 13 is a division answer, not a number of rows.
A baby elephant weighs 264 pounds. Its mother weighs 15 times as much. What does the mother weigh?
3,960 pounds
264 × 10 = 2,640.
264 × 5 = 1,320.
2,640 + 1,320 = 3,960.
279 is 264 + 15, reading "times as much" as "more than". 2,640 and 1,320 are the two partial products on their own; the answer needs both.
What is the product of 8 and 6?
8 × 6 = 48.
Now add 7 to that product.
48 + 7 = 55.
As one expression: (8 × 6) + 7. The product is grouped, then 7 is added.
The order you work in is fixed, whatever order the symbols appear in. Find 3 × (12 − 4) + 2.
Parentheses first. 12 − 4 = 8.
Multiply next. 3 × 8 = 24.
Add last. 24 + 2 = 26.
Writing an expression from a story is the same skill backwards. A carton holds 6 boxes with 8 pens in each box. Priya takes out 5 broken pens and then gives away 10 pens. How many are left?
Pens in the carton: 6 × 8.
Taken out: 5, then 10.
The expression is 6 × 8 − 5 − 10, or (6 × 8) − (5 + 10). Both say the same thing.
Parentheses first. Then multiply and divide, left to right. Then add and subtract, left to right.
"The product of" and "the sum of" are groups. Finish the group before you do anything else to it.
Reading left to right. 7 + 8 × 6 feels like 15 × 6, which is 90. It is 7 + 48, which is 55. We read sentences left to right, so this is the natural mistake. The multiplication waits for nobody.
In a story, check what a subtraction is taken from. "Took out 5 and gave away 10" are both taken from the total, so 6 × 8 − (10 − 5) is wrong.
Practice
What is the value of (9 − 3) × 4?
24
Parentheses first. 9 − 3 = 6.
6 × 4 = 24.
Devon added 5 to the product of 7 and 9. Which expression shows what Devon did?
(7 × 9) + 5
"The product of 7 and 9" is the group 7 × 9. Then 5 is added to it.
7 × (9 + 5) adds the 5 to 9 before multiplying. (7 + 9) × 5 multiplies a sum by 5. 7 × 9 × 5 never adds.
What is the value of 40 − 6 × 5 + 2?
12
Multiply first. 6 × 5 = 30.
Then left to right. 40 − 30 = 10.
10 + 2 = 12.
172 comes from working left to right: 34 × 5 + 2. 8 comes from adding the 2 to the 30 before subtracting. 238 comes from treating both ends as groups, (40 − 6) × (5 + 2).
A tray holds 4 rows of 6 muffins. Rosa bakes 3 full trays. She gives away 10 muffins and eats 2. Which expression shows how many muffins are left?
3 × (4 × 6) − 10 − 2
One tray is 4 × 6. Three trays is 3 × (4 × 6). Then 10 and 2 are both taken away from that total.
3 × 4 × 6 − (10 − 2) only takes away 8. 3 × (4 × 6 − 10 − 2) takes 12 muffins off each tray, which is 36 in all. 4 × 6 − 10 − 2 × 3 multiplies only the 2 by 3.
Mixed check 2
These two questions do not say which skill they use. Decide the method first, then answer.
What is the value of 50 − (8 + 7) × 2?
Method: Order of operations
20
Parentheses first. 8 + 7 = 15.
Multiply. 15 × 2 = 30.
Subtract. 50 − 30 = 20.
70 subtracts first and then doubles. 98 ignores the parentheses and works left to right. 56 drops the parentheses and only doubles the 7.
A ferry carries 35 cars on each trip. How many trips are needed to move 912 cars?
Method: Multiply and divide whole numbers
27 trips
912 ÷ 35 is 26 with 2 left over.
26 trips carry 910 cars. The last 2 cars need one more trip.
27 trips are needed.
26 drops the remainder. This is a whole-number division with a "needed" in the question, so the leftover counts.
Which number makes 5 × ? = 35 true?
5 × 7 = 35, so the missing number is 7.
Another way to see it: 35 ÷ 5 = 7. Dividing undoes multiplying.
Now a story with a letter in it. A florist earned $65 on Saturday. Roses brought in $26 and the rest came from tulips at $3 each. She wrote 26 + 3 × t = 65 to find t, the number of tulips sold.
Undo the adding first. 3 × t = 65 − 26.
3 × t = 39.
Undo the multiplying. t = 39 ÷ 3.
t = 13. She sold 13 tulips.
Some stories need two steps before the letter appears. A stage has a perimeter of 120 feet. Rope-light strands are 8 feet long and cost $15 each. Which equation finds b, the cost of the lights?
Strands needed: 120 ÷ 8.
Cost: that many strands times 15.
b = (120 ÷ 8) × 15.
A strip diagram shows the same thing as a picture: a bar for the total, cut into equal parts for each unknown.
Find the equation that says the same thing as the story, one phrase at a time. Then undo one operation at a time, working from the total back.
Right numbers, wrong order. Every option in a "which equation" question uses the numbers from the story. The wrong ones divide by the wrong thing or multiply too early. Test each option against the story: "120 ÷ 8 is the number of panels" is a sentence you can check.
When solving, do not stop after the first undo. 65 − 26 = 39 is the money from tulips, not the number of tulips.
Practice
Solve 9 × m = 72.
m = 8.
72 ÷ 9 = 8.
A stage has a perimeter of 96 feet. Rope-light strands are 6 feet long and cost $20 each. Which equation finds c, the total cost of the lights?
c = (96 ÷ 6) × 20.
Strands needed: 96 ÷ 6 = 16.
Cost: 16 × 20 = 320.
The lights cost $320.
c = (96 ÷ 20) × 6 divides the perimeter by the price. c = 96 × 6 × 20 multiplies everything. c = 96 ÷ (6 × 20) divides by the cost of one foot... except 6 × 20 is not that either.
Jada bought a backpack for $18 and 4 notebooks that each cost the same amount. She spent $30 in all. She wrote 18 + 4 × n = 30 to find n, the cost of one notebook. What is n?
n = 3.
Undo the adding. 4 × n = 30 − 18.
4 × n = 12.
Undo the multiplying. n = 12 ÷ 4.
n = 3. Each notebook cost $3.
12 is the money spent on notebooks, not the price of one. 7.50 comes from dividing 30 by 4 without taking off the backpack first.
The strip diagram shows a total of 135 split into equal parts.
Which equation matches the diagram, and what is p?
Equation: 5 × p = 135. p = 27
Five equal parts make 135, so 5 × p = 135.
p = 135 ÷ 5.
p = 27.
135 × 5 = p would make p bigger than the total. p + 5 = 135 treats the 5 parts as 5 single units added on. p ÷ 5 = 135 has the total and the part swapped.
What is 1/4 + 1/4?
Two quarters. 1/4 + 1/4 = 2/4, which is the same as 1/2.
The pieces are the same size, so you only count the pieces. The denominator stays 4.
When the pieces are different sizes, make them the same size first. Find 1/2 + 1/3.
Halves and thirds both fit into sixths. 1/2 = 3/6. And 1/3 = 2/6.
3/6 + 2/6 = 5/6.
Multiplying a fraction by a whole number: 3/4 × 5 means five lots of three quarters.
3/4 × 5 = 15/4.
15/4 is 3 wholes and 3/4 more, so 3 3/4.
Dividing has two shapes, and they are one word apart.
1/3 ÷ 4 means a third shared among 4. Each share is a smaller piece: 1/12.
4 ÷ 1/3 means how many thirds fit in 4 wholes. Each whole has 3 thirds: 12.
Add or subtract: same-size pieces first, then count the pieces.
A unit fraction divided by a whole number gets smaller: multiply the denominator.
A whole number divided by a unit fraction gets bigger: count the pieces, multiply.
Adding tops and bottoms. 1/2 + 1/3 is not 2/5. You cannot count halves and thirds together until they are the same kind of piece.
The two divisions. "1/3 pound divided into 4 parts" and "4 pounds divided into 1/3-pound parts" use the same numbers and give 1/12 and 12. Ask: is the answer a smaller piece, or a count of pieces?
Practice
What is 3/8 + 2/8?
5/8
Same-size pieces, so count them. 3 eighths plus 2 eighths is 5 eighths.
5/16 would come from adding the denominators too.
The shaded part of the bar shows a fraction. The crossed-out parts were subtracted from it.
Which expression does the model show, and what is the result?
Expression: 5/6 − 2/6. Result: 3/6, which is 1/2.
The bar has 6 parts, 5 are shaded, 2 of the shaded parts are crossed out. Start with 5/6, subtract 2/6, and 3 shaded parts remain.
5/6 + 2/6 adds instead of taking away. 5/6 − 2/5 counts the crossed parts out of the shaded parts instead of out of the whole bar. 6/6 − 2/6 starts from the whole bar instead of the shaded part.
What is 2/3 + 1/4?
11/12
Thirds and quarters both fit into twelfths. 2/3 = 8/12. And 1/4 = 3/12.
8/12 + 3/12 = 11/12.
3/7 comes from adding tops and bottoms.
Mr. Okafor has 1/2 pound of trail mix divided into 5 equal parts. What is the size of each part? Then: how many 1/2-pound bags can be filled from 5 pounds?
Each part: 1/10 pound. Bags from 5 pounds: 10.
1/2 ÷ 5: half a pound shared 5 ways is a smaller piece, 1/10 pound.
5 ÷ 1/2: each pound fills 2 half-pound bags, so 5 pounds fill 10.
10 pounds is the answer to the other question. 5/2 pounds multiplies instead of dividing. 2/5 pound has the numbers upside down.
Mixed check 3
These two questions do not say which skill they use. Decide the method first, then answer.
A ribbon 1/4 yard long is cut into 3 equal pieces. How long is each piece?
Method: Fractions
1/12 yard
A quarter shared 3 ways is a smaller piece. 1/4 ÷ 3 = 1/12.
12 yards answers "how many 1/4-yard pieces fit in 3 yards", the other division. 3/4 yard multiplies. 1/7 yard adds the 3 to the denominator.
A pool deck has a perimeter of 150 feet. Safety rope comes in 10-foot sections that cost $4 each. Which equation finds r, the total cost of rope for the whole perimeter?
Method: Equations from a story
r = (150 ÷ 10) × 4.
Sections needed: 150 ÷ 10 = 15.
Cost: 15 × 4 = 60.
The rope costs $60.
r = (150 ÷ 4) × 10 divides by the price instead of the section length. The other two never find the number of sections.
How many unit squares cover a rectangle that is 5 units long and 3 units wide?
3 rows of 5 squares. 5 × 3 = 15 square units.
Area uses two dimensions.
Volume is that same layer, stacked. A box has a bottom layer of 5 by 3 cubes and is 4 cubes tall.
One layer: 5 × 3 = 15 cubes.
Four layers: 15 × 4 = 60 cubes.
The volume is 60 cubic units. The formula says the same thing: V = l × w × h.
Area of one face uses only the two dimensions of that face. The floor of this box is 5 by 3, so its area is 15 square units, whatever the height.
Perimeter is the distance around a flat shape. A rectangle 2.5 meters wide and 1.2 meters tall has perimeter 2 × (2.5 + 1.2), which is 7.4 meters.
Volume is one layer times the number of layers: length × width × height.
Area of a face uses two dimensions. Perimeter adds up the sides.
Using all three numbers. A box question gives you three lengths, so it feels wasteful not to use them all. If the question asks for the area of the floor, only two of them belong. If it asks for the volume, all three multiply. Nothing in this section ever adds the three.
Practice
The bottom layer of a box is 6 cubes long and 4 cubes wide. How many cubes are in that one layer?
24
6 × 4 = 24.
10 comes from adding the two numbers.
A rectangular prism is 6 units long, 4 units wide and 5 units tall.
Expression: 6 × 4 × 5. Volume: 120 cubic units.
One layer: 6 × 4 = 24.
Five layers: 24 × 5 = 120.
6 + 4 + 5 adds the edges. 6 × 4 + 5 adds the height instead of stacking that many layers. 2 × (6 + 4) is a perimeter.
A storage crate is shaped like the prism shown. The floor is shaded.
What is the area of the shaded floor?
24 square inches
The floor is 8 inches by 3 inches.
8 × 3 = 24.
96 square inches is the volume, using all three numbers. 32 square inches is the front face, 8 by 4. 15 square inches adds the three edges.
A rectangular sign is 2.5 meters wide and 1.2 meters tall. What is the perimeter of the sign in meters?
7.4 meters
2.5 + 1.2 = 3.7.
2 × 3.7 = 7.4.
3.7 is only two of the four sides. 3 is the area, 2.5 × 1.2.
To plot the point (3, 5), how far across do you go, and how far up?
Across 3, then up 5. The first number is always the across number (x). The second is the up number (y).
The point where the two axes cross is the origin, (0, 0). Across 0, up 0.
A rule turns a list of x-values into points. Compare two rules.
y = x + 3 gives (0, 3), (1, 4), (2, 5). The line starts 3 above the origin and climbs 1 each step.
y = 3 × x gives (0, 0), (1, 3), (2, 6). The line passes through the origin and climbs 3 each step.
To check whether a table matches an equation, put each x into the equation and see if you get that table's y. One pair that fails rules the table out.
Across first, then up.
A rule with + starts above the origin and climbs 1 each step. A rule with × passes through the origin and climbs by the multiplier each step.
Swapping the two numbers. Reading (3, 5) as up 3, across 5 puts the point in the wrong place. "Up" feels like the first thing to do. It is the second.
A story that starts with an amount already there ("was 2 inches tall when planted") is an adding rule, not a multiplying rule, even if the growth each week is the number that catches your eye.
Practice
Point A is plotted on the grid.
What are the coordinates of point A?
(4, 2)
Point A is 4 across and 2 up.
(2, 4) swaps the two numbers.
Which ordered pair names the origin, where the x-axis and y-axis cross?
(0, 0)
At the crossing point you have gone 0 across and 0 up.
(1, 0) and (0, 1) are one step along an axis. (1, 1) is one step along both.
Which set of points follows the rule y = x + 4?
(1, 5), (2, 6), (3, 7)
Put each x in. 1 + 4 = 5. 2 + 4 = 6. 3 + 4 = 7.
(1, 4), (2, 8), (3, 12) follows y = 4 × x. (1, 4), (2, 5), (3, 6) follows y = x + 3. (1, 1), (2, 2), (3, 3) follows y = x.
A table shows the points (1, 3), (2, 6), (3, 9).
Rule: y = 3 × x. Through the origin: Yes.
Each y is three times its x. 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9.
A multiplying rule gives (0, 0), so the line passes through the origin.
y = x + 2 works for the first point only. y = x + 3 is the trap: the first y is 3, but 2 + 3 is 5, not 6. y = x × x gives 1, 4, 9.
Mixed check 4
These two questions do not say which skill they use. Decide the method first, then answer.
A toy chest is 30 inches long, 20 inches wide and 18 inches tall. What is its volume in cubic inches?
Method: Volume and area
10,800 cubic inches
One layer: 30 × 20 = 600.
Eighteen layers: 600 × 18 = 10,800.
600 is the floor area only. 68 adds the three edges. The word "volume" and the unit "cubic" tell you all three numbers multiply.
A day on Planet Kell lasts 41.6 hours. A day on Planet Vos lasts 6.9 hours. About how many times as long is a day on Kell as a day on Vos?
Method: Decimal operations
About 6 times
Round to friendly numbers. 41.6 is about 42 and 6.9 is about 7.
42 ÷ 7 = 6.
About 48 times adds the two numbers. About 35 times subtracts them. About 7 times is the divisor itself.
Does a square have four equal sides? Does a rhombus?
Yes and yes. A rhombus is any quadrilateral with four equal sides. A square has four equal sides, so every square is also a rhombus.
A shape belongs to every group whose definition it meets.
The whole family, from biggest group to smallest: quadrilateral, then parallelogram (two pairs of parallel sides), then rectangle (four right angles) and rhombus (four equal sides), then square (both). A trapezoid has exactly one pair of parallel sides, so it is a quadrilateral but not a parallelogram.
Numbers sort the same way. A prime number has exactly two factors, 1 and itself. 2 has factors 1 and 2, so 2 is prime even though it is even.
Reading a stem-and-leaf plot: the stem is the tens digit, each leaf is a ones digit. In the row 2 | 0 3 3 7, the values are 20, 23, 23 and 27. In a dot plot, each dot is one item at that value.
Money words: sales tax is added to a price when you buy something. Income tax is taken from money you earn. Property tax is paid on land or a house you own. Payroll tax is taken from a paycheck to fund Social Security and Medicare.
A shape belongs to every group whose definition it meets. A prime number has exactly two factors. In a stem-and-leaf plot, read the stem and the leaf together as one number.
"All even numbers are composite." It is true for every even number except 2, and 2 is the one the test asks about.
Reading a leaf alone. The leaf 7 in the row with stem 2 is 27, not 7. The greatest and least values in a plot come from the first and last rows, read with their stems.
Practice
Which of these numbers is prime?
23
23 has exactly two factors, 1 and 23.
21 is 3 × 7. 27 is 3 × 9. 25 is 5 × 5. Each of those has more than two factors.
A student said that 2 is a composite number because it is even. Is the student right?
No, 2 has exactly two factors, so it is prime
The factors of 2 are 1 and 2. Exactly two factors is the definition of prime. Being even has nothing to do with it.
"Every even number is composite" is true from 4 upwards and false at 2. "Only one factor" describes 1, not 2. "Neither" also describes 1.
The stem-and-leaf plot shows the number of pages nine students read in one evening.
| Stem | Leaves |
|---|---|
| 1 | 4 8 |
| 2 | 0 3 3 7 |
| 3 | 1 5 |
Key: 2 | 3 means 23
What is the difference between the greatest number of pages and the least?
21
Least: stem 1, leaf 4, so 14. Greatest: stem 3, leaf 5, so 35.
35 − 14 = 21.
2 comes from subtracting the stems only. 1 comes from subtracting the leaves only.
Which statement is true?
Every square is a rectangle
A rectangle is a quadrilateral with four right angles. A square has four right angles, so it meets the definition.
A rectangle that is 3 by 5 is not a square, so not every rectangle is a square. A trapezoid has only one pair of parallel sides, so it is never a parallelogram. A rhombus with slanted corners is not a square.
Mixed run
Every skill in this guide, in no order, and nothing tells you which one to use. Name the method on your paper before you work it. That is exactly what the test asks.
Money added to the price of a video game when you buy it at a store is called what?
Method: Shapes and data
Sales tax
Sales tax is added at the moment of buying. Income tax comes out of earnings. Property tax is on land or a house. Payroll tax comes out of a paycheck for Social Security and Medicare.
A shoebox is 9 cm long, 5 cm wide and 7 cm tall. What is the area of its 9 cm by 5 cm floor?
Method: Volume and area
45 square centimeters
The floor uses two dimensions. 9 × 5 = 45.
315 is the volume. See the next question.
The same shoebox is 9 cm long, 5 cm wide and 7 cm tall. What is its volume?
Method: Volume and area
315 cubic centimeters
One layer: 9 × 5 = 45.
Seven layers: 45 × 7 = 315.
Same box as the question before. One word, "volume" instead of "area", brings in the third number.
A baker has 3 pounds of dough and makes rolls that each use 1/5 pound. How many rolls can she make?
Method: Fractions
15 rolls
3 ÷ 1/5 asks how many fifths are in 3 wholes. Each pound has 5 fifths.
3 × 5 = 15.
1/15 is the answer to the next question, where the fraction is the thing being shared.
A baker shares 1/5 pound of dough equally among 3 rolls. How much dough is in each roll?
Method: Fractions
1/15 pound
1/5 ÷ 3 shares a fifth into 3 smaller pieces. Each piece is a fifteenth.
15 is the answer to the question before this one. Same numbers, the other way round.
Which ordered pair lies on the line for y = x + 5?
Method: Coordinate plane
(2, 7)
Put x = 2 into the rule. 2 + 5 = 7.
(7, 2) swaps across and up. (2, 10) uses y = 5 × x. (5, 0) is on the x-axis, not on this line.
Luis bought a $22 ticket and 3 snacks that each cost the same amount. He spent $34 in all. He wrote 22 + 3 × s = 34. What is s?
Method: Equations from a story
s = 4.
Undo the adding. 3 × s = 34 − 22.
3 × s = 12.
Undo the multiplying. s = 12 ÷ 3.
s = 4. Each snack cost $4.
Amara subtracted 4 from the product of 6 and 9. What number did she get?
Method: Order of operations
50
The product of 6 and 9 is the group. 6 × 9 = 54.
54 − 4 = 50.
30 comes from subtracting 4 from 9 first and then multiplying by 6.
Which list shows the numbers in order from greatest to least?
Method: Compare and round decimals
4.5, 4.35, 4.305, 4.3
Write them all in thousandths: 4.500, 4.350, 4.305, 4.300. Those go down at every step.
4.3, 4.305, 4.35, 4.5 is least to greatest. The list beginning 4.5, 4.305 puts 305 before 350, which is the "longer looks bigger" slip.
A 12.6-pound bag of rice is poured equally into 7 containers. How many pounds of rice are in each container?
Method: Decimal operations
1.8 pounds
Treat 12.6 as 126 and divide.
126 ÷ 7 = 18.
Put the point back one place. 12.6 ÷ 7 = 1.8.
18 forgets the point.
A warehouse packs 1,470 jars equally into 42 crates. How many jars go in each crate?
Method: Multiply and divide whole numbers
35 jars
Estimate first: 1,470 ÷ 42 is close to 1,600 ÷ 40, so around 35 to 40.
42 goes into 147 three times. 42 × 3 = 126. 147 − 126 = 21.
Bring down the 0 to make 210. 42 × 5 = 210.
1,470 ÷ 42 = 35, with nothing left over.
Read this once, cold. The first move of each skill, then a few habits.
| Skill | First move |
|---|---|
| Compare and round decimals | Add zeros so every decimal has the same number of digits. To round, look at one digit to the right. |
| Decimal operations | Line up the points to add or subtract. Count decimal places in both factors to multiply. Divide as whole numbers, then put the point back. |
| Multiply and divide whole numbers | Estimate with a friendly divisor first. If the question says "needed", a remainder means one more. |
| Order of operations | Parentheses, then multiply and divide, then add and subtract. "The product of" is a group. |
| Equations from a story | Match the story phrase by phrase, then undo one operation at a time from the total back. |
| Fractions | Same-size pieces before adding. Fraction ÷ whole gets smaller; whole ÷ unit fraction counts pieces. |
| Volume and area | Volume is one layer times the layers. A face uses two numbers. Perimeter adds the sides. |
| Coordinate plane | Across first, then up. A + rule starts above the origin; a × rule passes through it. |
| Shapes and data | A shape belongs to every group it fits. Prime means exactly two factors. Read stem and leaf together. |