A university library's archival team is digitizing rare manuscripts using three scanner mo… — ACT

A university library's archival team is digitizing rare manuscripts using three scanner models: Model X, Model Y, and Model Z. The ratio of the scanning speed of Model X to Model Y is $3:5$ , and the ratio of the scanning speed of Model Y to Model Z is $2:3$ . If Model X and Model Z together scan a total of $420$ pages in $1$ hour, how many total pages will all three scanner models combined scan in $3$ hours?

A
620
B
1,240
1,860Answer
D
2,310
E
2,520

Answer

All three scanner models combined will scan 1,860 pages in 3 hours.

To find the total number of pages scanned, first unify the given ratios using Model Y as the common bridge: Model X to Model Y is

3:5

(

6:10

) and Model Y to Model Z is

2:3

(

10:15

), yielding an extended ratio of

6:10:15

. Since Model X and Model Z together scan

420

pages in

1

hour,

6k + 15k = 420

, which yields

k = 20

. The total hourly rate for all three models is

(6 + 10 + 15) \times 20 = 620

pages per hour. Over

3

hours, the total volume scanned is

620 \times 3 = 1,860

pages.

Step-by-Step Solution

Unify the two ratios into a single three-part ratio for Model X : Model Y : Model Z.

Model X : Model Y =

3 : 5 = 6 : 10

and Model Y : Model Z =

2 : 3 = 10 : 15

. Thus, Model X : Model Y : Model Z =

6 : 10 : 15

Model Y is the common element between both given ratios, so we scale both ratios so that Model Y has the same value (

10

) in both.

Set up an equation based on the combined hourly output of Model X and Model Z.

Let

k

be the multiplier. Rate of X

= 6k

, Rate of Z

= 15k

6k + 15k = 420 \implies 21k = 420 \implies k = 20

Model X and Model Z together produce

420

pages per hour, which allows us to solve for the constant ratio multiplier

k

Calculate the combined hourly rate for all three scanner models.

Combined Rate

= (6 + 10 + 15)k = 31k = 31 \times 20 = 620

pages per hour.

Summing the ratio parts of all three models gives the total rate per hour.

Multiply the combined hourly rate by 3 hours to find the total production.

620 \text{ pages/hour} \times 3 \text{ hours} = 1,860 \text{ pages}

The question asks for the total output over a 3-hour period.

Key Concept

Unifying compound ratios into a single extended ratio and scaling proportional rates over time.

Estimated Time:2m 0s

Question

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Step-by-Step Solution

Key Concept

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