Practice 1: Solving Word Problems
(1)
Factory Production
Question: Factory A produces 326 dresses a day. Factory B produces
107 more dresses a day than Factory A. (a) How many dresses does Factory B
produce in a day? (b) How many dresses do the two factories produce in a day?
Solving
Way:
·
Understanding
the problem: We need to find out how many
dresses Factory B makes, and then the total dresses made by both factories.
·
(a) Dresses
from Factory B:
o Factory A makes 326 dresses.
o Factory B makes 107 more than Factory A. This means
we add 107 to Factory A's production.
o Dresses from Factory B = Dresses from Factory A + 107
o Dresses from Factory B = 326 + 107 = 433 dresses
o Answer (a): Factory B produces 433 dresses in a day.
·
(b) Total
dresses from both factories:
o To find the total, we add the dresses from Factory A and
Factory B.
o Total dresses = Dresses from Factory A + Dresses from
Factory B
o Total dresses = 326 + 433 = 759 dresses
o Answer (b): The two factories produce 759 dresses in a day.
(2)
Chair Purchase
Question: Munah bought 4 similar chairs at $384. Rehna bought 9 such
chairs from the same shop. How much did Rehna pay for the 9 chairs?
Solving
Way:
·
Understanding
the problem: First, we need to find the price of
one chair. Since all chairs are "similar" and from the "same
shop," they cost the same amount. Then, we can calculate the cost for 9
chairs.
·
Step 1: Find
the cost of one chair.
o Munah bought 4 chairs for $384.
o Cost of 1 chair = Total cost / Number of chairs
o Cost of 1 chair = $384 / 4 = $96
o So, each chair costs $96.
·
Step 2: Find
the cost for Rehna's 9 chairs.
o Rehna bought 9 chairs.
o Cost for Rehna = Number of chairs Rehna bought × Cost of 1
chair
o Cost for Rehna = 9 × $96 = $864
o Answer: Rehna paid $864 for the 9 chairs.
(3)
Wai Keong's Salary and Savings
Question: Wai Keong's monthly salary is twice the amount he saved in
April. He saved $3500 in April. In May, he saved $4200 less than the amount he
spent. (a) How much is his monthly salary? (b) How much did he save in May?
Solving
Way:
·
Understanding
the problem: This problem has two parts. First,
we calculate Wai Keong's salary based on his April savings. Second, we figure
out his May savings based on a comparison with his May spending (though the
information about spending in May seems a bit tricky, let's re-read carefully).
·
(a) Wai
Keong's monthly salary:
o He saved $3500 in April.
o His monthly salary is twice the amount he saved in
April.
o Monthly salary = 2 × April savings
o Monthly salary = 2 × $3500 = $7000
o Answer (a): His monthly salary is $7000.
·
(b) How much
did he save in May?
o The question states: "In May, he saved $4200 less than
the amount he spent." This sentence is a bit tricky without knowing how
much he spent in May. It implies we don't know his May spending.
o Let's assume there might be a typo in the original question
or it's a trick question for part (b) as stated. If we are meant to calculate
his May savings, we need more information about his May spending.
o Correction/Clarification Needed: If the question meant "He saved $4200 in
May" (which is common phrasing for saving), then it's straightforward. Or
if it meant "He saved $4200 less than his salary in May."
o Given the exact phrasing, we cannot determine his May
savings without knowing his May spending.
o However, if the question intends to say he spent
$4200 more than he saved in May, or some other relationship, it's
ambiguous.
o Let's assume the question implies a relationship between his
saving and spending, and perhaps it meant "he saved $4200 in May" (a
common scenario where savings are given directly) or "he spent $4200 less
than his salary" which then could relate to savings. But as written,
"saved $4200 less than the amount he spent" means: Saved = Spent - $4200.
We don't know Spent. So, this part cannot be solved with the given information.
o If the intent was that he saved a specific amount or a
difference from his salary, the question would be phrased differently.
o Conclusion for (b): With the given information, we cannot
determine how much he saved in May.
o Self-correction based on common math problem patterns: Sometimes, these problems implicitly assume a certain
income (like his salary) and then relate savings/spending to that. If he earned
$7000 (his salary) and spent an amount, his savings would be Salary -
Spent. If "saved $4200 less than the amount he spent" is the only
information, and no other relation to his salary or total income is given for
May, then it's unsolvable. Let's consider if it could be a typo and it
meant something like "he saved $4200 in May" or "he spent $4200
in May, and saved the rest". But based strictly on the sentence,
it's problematic.
Let's assume a common mistake in question formulation and
assume it should be related to his salary or April savings in a different way. If the question implied that his spending was $4200 more
than his savings in May (which is "saved $4200 less than spent"), and
we don't know either, it's a system with two unknowns and one equation. Thus,
it's unsolvable as written.
I will indicate this ambiguity in the answer.
o Answer (b): The information provided ("In May, he saved
$4200 less than the amount he spent") is insufficient to determine how
much he saved in May, as his spending in May is unknown.
(4)
Amy and Mia's Savings
Question: Amy saved $3210. Mia saved twice as much as Amy. Amy saved
3 times as much as Ned. (a) How much did Ned save? (b) How much money must Mia
give to Ned so that they each have the same amount of money?
Solving
Way:
·
Understanding
the problem: We have three people: Amy, Mia, and
Ned. We know Amy's savings, and how Mia's and Ned's savings relate to Amy's. We
need to find Ned's savings, and then how to equalize Mia's and Ned's money.
·
Information
given:
o Amy's savings = $3210
o Mia's savings = 2 × Amy's savings
o Amy's savings = 3 × Ned's savings
·
(a) How much
did Ned save?
o We know Amy saved 3 times as much as Ned.
o So, Ned's savings = Amy's savings / 3
o Ned's savings = $3210 / 3 = $1070
o Answer (a): Ned saved $1070.
·
(b) How much
money must Mia give to Ned so that they each have the same amount of money?
o Step 1: Find Mia's savings.
§ Mia saved twice as much as Amy.
§ Mia's savings = 2 × $3210 = $6420
o Step 2: Understand the goal. Mia and Ned need to have the same amount. This means they
should each have (Mia's savings + Ned's savings) / 2.
o Total money for Mia and Ned = Mia's savings + Ned's savings
o Total money = $6420 + $1070 = $7490
o Amount each should have = Total money / 2
o Amount each should have = $7490 / 2 = $3745
o Step 3: Calculate how much Mia needs to give Ned.
§ Mia currently has $6420 and needs to have $3745.
§ Amount Mia gives away = Mia's current savings - Amount she
should have
§ Amount Mia gives away = $6420 - $3745 = $2675
§ (Check: If Ned gets $2675, he will have $1070 + $2675 =
$3745, which is correct.)
o Answer (b): Mia must give $2675 to Ned.
(5)
Mr. Wong's Air Tickets
Question: Mr. Wong bought 6 air tickets. 4 of them cost $1096 each
and the others cost $1487 each. How much did he spend altogether?
Solving
Way:
·
Understanding
the problem: Mr. Wong bought two types of
tickets with different prices. We need to calculate the cost for each type and then
add them up for the total.
·
Step 1:
Calculate the cost of the first type of tickets.
o Number of tickets = 4
o Cost per ticket = $1096
o Total cost for first type = 4 × $1096 = $4384
·
Step 2:
Calculate the number of the second type of tickets.
o Total tickets bought = 6
o First type of tickets = 4
o Number of second type = Total tickets - First type tickets
o Number of second type = 6 - 4 = 2 tickets
·
Step 3:
Calculate the cost of the second type of tickets.
o Number of tickets = 2
o Cost per ticket = $1487
o Total cost for second type = 2 × $1487 = $2974
·
Step 4:
Calculate the total amount spent.
o Total spent = Cost of first type + Cost of second type
o Total spent = $4384 + $2974 = $7358
o Answer: He spent $7358 altogether.
(6)
Concert Attendees
Question: There were 6872 men and women at a concert. There were 2150
more men than women. How many men were there at the concert?
Solving
Way:
·
Understanding
the problem: We know the total number of people
and the difference between the number of men and women. We need to find the
exact number of men.
·
Let's use
variables (or think in terms of parts):
o Let W = number of women
o Let M = number of men
·
From the
problem:
o M+W=6872 (Total people)
o M=W+2150 (2150 more men than women)
·
Step 1:
Substitute to solve for W (women).
o Replace M in the first equation with (W+2150): (W+2150)+W=6872
o Combine the W's: 2W+2150=6872
o Subtract 2150 from both sides: 2W=6872−2150 2W=4722
o Divide by 2 to find W: W=4722/2=2361 women
·
Step 2: Find
the number of men (M).
o Now that we know W, we can use either equation. Let's use M=W+2150.
o M=2361+2150=4511 men
·
Check: Does M+W=6872? 4511+2361=6872. Yes, it matches.
·
Answer:
There were 4511 men at the concert.
o Alternative thinking (without explicit algebra):
§ If there were an equal number of men and women, the total
would be 6872.
§ But men are 2150 more.
§ Take away the "extra" men first: 6872−2150=4722.
§ Now, this 4722 represents the total if men and women were
equal. So, divide by 2 to find the number of women: 4722/2=2361 women.
§ Add the "extra" men back to the women to get the
number of men: 2361+2150=4511 men.
(7)
Josephine's Biscuits
Question: Josephine bought 8 packets of biscuits. Each packet had a
mass of 500 g. She repacked them into 3 boxes. The first box was twice as heavy
as the second box. The third box was 60 g heavier than the second box. What was
the mass of the second box?
Solving
Way:
·
Understanding
the problem: First, find the total mass of all
biscuits. Then, distribute this total mass into three boxes based on their
relative weights, and find the mass of the second box.
·
Step 1:
Calculate the total mass of biscuits.
o Number of packets = 8
o Mass per packet = 500 g
o Total mass = 8 packets × 500 g/packet = 4000 g
·
Step 2:
Define variables for the mass of each box.
o Let B2 = mass of the second box (in g)
o Mass of first box = 2×B2 (twice as heavy as the second box)
o Mass of third box = B2+60 (60 g heavier than the second box)
·
Step 3: Set
up an equation for the total mass.
o The sum of the masses of the three boxes must equal the
total mass of biscuits.
o Mass of first box + Mass of second box + Mass of third box =
Total mass
o (2×B2)+B2+(B2+60)=4000
·
Step 4:
Solve the equation for B2.
o Combine the B2 terms: 2B2+B2+B2+60=4000 4B2+60=4000
o Subtract 60 from both sides: 4B2=4000−60 4B2=3940
o Divide by 4: B2=3940/4=985 g
·
Check
(optional but good practice):
o Second box (B2) = 985 g
o First box (2×B2) = 2×985=1970 g
o Third box (B2+60) = 985+60=1045 g
o Total = 1970+985+1045=4000 g. This matches!
·
Answer: The
mass of the second box was 985 g.
(8)
Alex, Gil, and Meg's Cards
Question: Alex, Gil, and Meg shared 740 cards. Alex had 20 fewer
cards than Gil and 120 fewer cards than Meg. How many cards did Alex have?
Solving
Way:
·
Understanding
the problem: We know the total number of cards
and how Alex's cards relate to Gil's and Meg's. We need to find the number of
cards Alex has. It's easiest to express everyone's cards in terms of Alex's
cards.
·
Define
variables:
o Let A = number of cards Alex had
o Let G = number of cards Gil had
o Let M = number of cards Meg had
·
From the
problem:
o A+G+M=740 (Total cards)
o "Alex had 20 fewer cards than Gil": A=G−20
§ This means Gil had 20 more than Alex: G=A+20
o "Alex had 120 fewer cards than Meg": A=M−120
§ This means Meg had 120 more than Alex: M=A+120
·
Step 1:
Substitute to express everything in terms of A.
o Substitute the expressions for G and M into the total cards
equation: A+(A+20)+(A+120)=740
·
Step 2:
Solve for A.
o Combine the A terms: A+A+A+20+120=740 3A+140=740
o Subtract 140 from both sides: 3A=740−140 3A=600
o Divide by 3: A=600/3=200 cards
·
Check
(optional):
o Alex (A) = 200 cards
o Gil (A+20) = 200+20=220 cards
o Meg (A+120) = 200+120=320 cards
o Total = 200+220+320=740 cards. This matches!
·
Answer: Alex
had 200 cards.
(9)
Fionna, Anuar, and Rizza's Cards
Question: Fionna, Anuar and Rizza had some cards. Fionna and Anuar
had a total of 349 cards. Fionna had 145 cards. Anuar had 3 times as many cards
as Rizza. How many cards did Fionna and Rizza have altogether?
Solving
Way:
·
Understanding
the problem: We are given several pieces of
information about the number of cards Fionna, Anuar, and Rizza have. We need to
find the total cards for Fionna and Rizza.
·
Information
given:
o Fionna + Anuar = 349 cards
o Fionna = 145 cards
o Anuar = 3 × Rizza cards
·
Step 1: Find
Anuar's cards.
o We know Fionna + Anuar = 349 and Fionna = 145.
o Anuar = (Fionna + Anuar) - Fionna
o Anuar = 349 - 145 = 204 cards
·
Step 2: Find
Rizza's cards.
o We know Anuar had 3 times as many cards as Rizza.
o So, Rizza's cards = Anuar's cards / 3
o Rizza's cards = 204 / 3 = 68 cards
·
Step 3:
Calculate the total cards for Fionna and Rizza.
o Total = Fionna's cards + Rizza's cards
o Total = 145 + 68 = 213 cards
·
Answer:
Fionna and Rizza had 213 cards altogether.
){kind=link}
0 Comments