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The phrase "a higher rate with" is correct and usable in written English.
It can be used when discussing a comparison of rates, such as interest rates, prices, or fees associated with a service or product.
Example: "The new plan offers a higher rate with additional benefits compared to the previous one."
Alternatives: "an increased rate alongside" or "a superior rate in conjunction with".
Exact(17)
This portfolio returns at a higher rate with less risk using the simple concept of diversification.
But, as Kleinbard points out, the plan would end up looking much like a new payroll tax levied at a higher rate, with no upper limit on the income it applies to.
In this case, the client is likely to have to refinance its equipment at a higher rate with a lender that will take a lien only on the equipment and allow the business to proceed with its expansion.
These estimates are similar, with overlapping confidence intervals for the effect of aging, but a higher rate with lithium exposure.
A comparison of occurrence by diagnoses in the non-traumatic disorders showed a higher rate with lumbar (27.7%), tendon (23.2%), and joint disorders (16.1%).
Beginning in tax year 2002, the state of Massachusetts gave taxpayers the option of checking a box on their 1040s to pay a higher rate, with the extra funds going to social services.
Similar(43)
If interest rates fall further, you remain locked into a high rate with a CD or an individual bond; you win with either one, right?
These were a high rate with 1,000 ml, a normal rate with 750 ml (Control) and low rate with 500 ml.
In a third set of injection scenarios, we show that starting injection at a high rate with ramping down, a low rate with ramping up or at a constant rate makes little difference to the overall injectivity of the reservoir.
Affeldt induces groundballs at a high rate with his sinking fastball and can also get the strikeout when the situation calls for it.
LM8 CTCs appeared faster, at a higher rate and with a greater number compared to Dunn CTCs.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com