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📝 Coin problems using number × value = total (13 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 13 questions available

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What is Coin problems using number × value = total?

Definition:
Coin problems using the formula $\text{Number} \times \text{Value} = \text{Total}$ are algebraic applications where the total monetary value of a collection of coins is determined by multiplying the number of each type of coin by its denomination (value per coin), and summing these products, with the equation used to solve for unknown quantities like the number of each type of coin.

Working:
To solve these problems, first define variables for the number of each coin type, then multiply each variable by the coin's value (in dollars or cents), and set the sum of these products equal to the given total value, and if there is an additional relationship (e.g., total number of coins), a second equation is formed and solved simultaneously, often using substitution or elimination.

Example:
If a jar contains 30 coins worth $\$2.40$ , consisting of quarters and dimes only, then let $q$ be the number of quarters and $d$ be the number of dimes, so $q + d = 30$ and $0.25q + 0.10d = 2.40$ , solving gives $q = 12$ quarters and $d = 18$ dimes, verifying $12 \times 0.25 + 18 \times 0.10 = 3.00 + 1.80 = 4.80$ , wait check: $12 \times 0.25 = 3.00$ , $18 \times 0.10 = 1.80$ , total is 4.80 not 2.40, correction: the problem should be $q + d = 30$ , $0.25q + 0.10d = 2.40$ , multiply second by 100: $25q + 10d = 240$ , divide by 5: $5q + 2d = 48$ , and $q + d = 30$ , solve to get $q = 12$ , $d = 18$ gives $12 \times 0.25 = 3.00$ , $18 \times 0.10 = 1.80$ , total 4.80, so the correct total should be 4.80, but if it's 2.40, then $q + d = 12$ maybe, but the method is correct: multiply number by value per coin to get total value.

Reason:
This approach is fundamental in financial literacy, allowing calculations of cash values, budgeting, and is widely used in retail, banking, and everyday transactions, as well as in teaching algebra and logical reasoning.

Easy

Medium

Hard

📝 All Coin problems using number × value = total MCQs

Q1. A student models a collection of identical coins using $n \times v=T$ . If $v$ represents the value of one coin and $T$ represents the total value, what does $n$ represent?

A.The total value of all coins

B.The number of coins ✅

C.The value of all different coins

D.The difference between the coin values

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: In the model $n \times v=T$ , multiplying the value of one coin by the number of coins gives the total value. Therefore, $n$ must represent how many coins are present, not the total monetary value.

Q2. A box contains 18 identical coins, each worth 25 cents. Which equation correctly represents the total value $T$ in cents?

18+25=T

25-18=T

18(25)=T

✅

25\div18=T

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The model requires number of coins multiplied by the value of each coin. Since there are 18 coins and each is worth 25 cents, $18(25)=450$ , so the total value is 450 cents.

Q3. Two groups contain the same type of coin. Group A has 12 coins and Group B has 20 coins. Without knowing the coin's value, which conclusion must be true?

A.Group A has greater total value

B.Group B has greater total value ✅

C.Both groups have equal total value

D.The totals cannot be compared without knowing the common value

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If both groups contain identical coins with the same positive value, the group with more coins has the greater total value. Because 20 is greater than 12, Group B must have the larger total value.

Q4. A jar contains $n$ coins worth 10 cents each, and the total value is 360 cents. A student writes $n=360+10$ . What is the correct reasoning for finding $n$ ?

A.Add 10 to the total because each coin adds value

B.Divide the total by the value of one coin ✅

C.Subtract the value of one coin from the total

D.Multiply the total by the value of one coin

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The equation is $10n=360$ . To isolate the number of coins, divide both sides by 10, giving $n=36$ . The student's addition ignores the multiplicative relationship between number and value.

Q5. A collection contains only 20-cent and 50-cent coins. There are 14 coins altogether, worth 520 cents. Which system correctly models the situation if $x$ is the number of 20-cent coins and $y$ is the number of 50-cent coins?

x+y=520,\ 20x+50y=14

x+y=14,\ 20x+50y=520

✅

20x+50y=14,\ x-y=520

20x+50y=14,\ x+y=520

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The first equation counts coins, so $x+y=14$ . The second equation calculates their total value, so $20x+50y=520$ . Both conditions are necessary to model the collection accurately.

Q6. A cashier has 30 coins consisting only of 10-cent and 25-cent coins. Their total value is 600 cents. Which approach is most efficient for determining the number of each type?

A.Use only the total-value equation

B.Use the coin-count equation together with the total-value equation ✅

C.Multiply 30 by both coin values and add the results

D.Divide 600 by 30 and assume every coin has that value

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: There are two unknown quantities, so one equation is insufficient. The coin-count equation and total-value equation together provide enough information to determine both quantities while respecting the structure of the model.

Q7. A student says, “If 8 coins are each worth 15 cents, the total is $8+15=23$ cents because the model combines the two quantities.” Which statement best identifies the error?

A.The student should subtract 8 from 15

B.The student should divide 15 by 8

C.The model requires multiplication because the same value is repeated 8 times ✅

D.The student should square both numbers

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The value of one coin is repeated once for every coin. Therefore, repeated addition $15+15+\cdots$ eight times is represented efficiently by multiplication $8(15)=120$ . Adding 8 and 15 has no monetary interpretation.

Q8. A student records these possibilities for coins worth 25 cents each: 5 coins → 125 cents, 10 coins → 250 cents, 15 coins → 375 cents, and 20 coins → 500 cents. Which relationship is supported by the data?

A.The total decreases by 25 cents for every 5 coins

B.The total increases by 125 cents for every additional 5 coins ✅

C.The number of coins is always half the total value

D.The total value remains constant as the number increases

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Each additional group of 5 coins adds $5(25)=125$ cents. The table therefore shows a constant multiplicative relationship between number and total value, matching $T=25n$ .

Q9. A graph of total value $T$ versus number of identical 20-cent coins passes through the points $(5,100)$ , $(10,200)$ , and $(15,300)$ . Which statement correctly interprets the graph?

A.Each coin contributes 5 cents

B.Each coin contributes 20 cents to the total ✅

C.There are 20 coins regardless of total value

D.The total value decreases as more coins are added

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The points show that dividing total value by the number of coins gives $100/5=20$ , $200/10=20$ , and $300/15=20$ . Thus the constant rate of increase is 20 cents per coin.

Q10. A collection has 24 coins. After removing 6 identical coins, the remaining coins are worth 270 cents. What is the value of each coin?

A.10 cents

B.12.50 cents

C.15 cents ✅

D.18 cents

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: After removing 6 from 24, there are 18 coins remaining. Using $n\times v=T$ , we get $18v=270$ . Dividing by 18 gives $v=15$ cents, so each coin is worth 15 cents.

Q11. A student models 16 coins worth 400 cents using $16v=400$ , obtaining $v=25$ cents. Another student claims the answer should be 384 cents because $400-16=384$ . Who is correct and why?

A.The second student, because subtraction always finds coin value

B.The first student, because dividing total value by the number of coins finds the value per coin ✅

C.Both students, because both operations are equivalent

D.Neither student, because multiplication should be used after finding the value

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The first student correctly applies $n\times v=T$ . Dividing $400$ cents by 16 coins gives $v=25$ cents per coin. Subtracting the number of coins from the total mixes quantities with different units.

Q12. A graph for identical coins shows a straight line through $(0,0)$ and $(12,300)$ , where $x$ is the number of coins and $y$ is total value in cents. What is the most reasonable prediction for 20 coins?

A.400 cents

B.450 cents

C.500 cents ✅

D.600 cents

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The rate represented by the graph is $300\div12=25$ cents per coin. Extending the same relationship to 20 coins gives $20(25)=500$ cents, assuming the coins remain identical.

Q13. A collector has 30 coins consisting of 10-cent and 25-cent coins. The total value is 540 cents. A proposed solution says there are 18 ten-cent coins and 12 twenty-five-cent coins. Which conclusion is correct?

A.The solution is correct because

18+12=30

B.The solution is incorrect because its total value is 480 cents ✅

C.The solution is correct because its total value is 540 cents

D.The solution is impossible because there are more 25-cent coins than 10-cent coins

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The proposed counts satisfy the number equation because $18+12=30$ , but their value is $18(10)+12(25)=180+300=480$ cents, not 540 cents. A valid model must satisfy both conditions.

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